Extended active disturbance rejection controller

ABSTRACT

Multiple designs, systems, methods and processes for controlling a system or plant using an extended active disturbance rejection control (ADRC) based controller are presented. The extended ADRC controller accepts sensor information from the plant. The sensor information is used in conjunction with an extended state observer in combination with a predictor that estimates and predicts the current state of the plant and a co-joined estimate of the system disturbances and system dynamics. The extended state observer estimates and predictions are used in conjunction with a control law that generates an input to the system based in part on the extended state observer estimates and predictions as well as a desired trajectory for the plant to follow.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a divisional of U.S. application Ser. No. 14/097,624filed Dec. 5, 2013, which is a continuation of U.S. application Ser. No.13/464,558 filed May 4, 2012, which is a continuation of U.S.application Ser. No. 12/195,353 filed Aug. 20, 2008, both of which arehereby incorporated by reference. This application claims the benefit ofU.S. Application No. 60/965,359 filed 20 Aug. 2007, which is herebyincorporated by reference. This application is a continuation in partand claims the benefit of the following applications: U.S. applicationSer. No. 12/067,141 filed 17 Mar. 2008; that claims priority to thefollowing: Patent Cooperation Treaty Appl. No. PCT/US2006/036156, filed18 Sep. 2006; U.S. Appl. No. 60/718,393, filed 19 Sep. 2005; U.S. Appl.No. 60/718,581, filed 19 Sep. 2005; U.S. Appl. No. 60/718,899, filed 20Sep. 2005; U.S. Appl. No. 60/728,928, filed 20 Oct. 2005; and U.S. Appl.No. 60/728,929 filed, 20 Oct. 2005; and U.S. application Ser. No.10/351,664 filed, 27 Jan. 2003 that claims priority to U.S. Appl. No.60/373,404, filed 18 Apr. 2002, all of which are hereby incorporated byreference.

This work was supported at least in part by NASA under NASA contractnumber GT3-52387. Accordingly, the United States government may havecertain rights herein.

TECHNICAL FIELD

The subject matter described herein relates to controllers, systems, andmethods for feedback control of various systems.

BACKGROUND

A feedback (closed-loop) control system 10, as shown in Prior Art FIG.1, is widely used to modify the behavior of a physical process, denotedas the plant 110, so it behaves in a specific desirable way over time.For example, it may be desirable to maintain the speed of a car on ahighway as close as possible to 60 miles per hour in spite of possiblehills or adverse wind; or it may be desirable to have an aircraft followa desired altitude, heading and velocity profile independently of windgusts; or it may be desirable to have the temperature and pressure in areactor vessel in a chemical process plant maintained at desired levels.All of these industrial tasks are accomplished today by usingtraditional feedback control, and the above are examples of whatautomatic control systems are designed to do, without humanintervention.

The key component in a feedback control system is the controller 120,which determines the difference between the output “y” of the plant 110,(e.g., the temperature or position) and its desired value, and producesa corresponding control signal “u” (e.g., modulating the power input toa heater or motor).

The goal of controller design is to make the difference between theactual output of the plant and the desired output as small as possibleas rapidly as possible. Today, controllers are employed in a largenumber of industrial control applications and in areas like robotics,aeronautics, astronautics, motors, motion control and thermal control,just to name a few.

INDUSTRIAL APPLICABILITY

The present system and method has potential applicability to a widerange of different industrial and commercial applications. The followingbrief synopsis is intended only to provide background on some exemplaryapplications to assist a person of ordinary skill in the art inunderstanding this disclosure more fully. Additional backgroundinformation on a select number of exemplary industrial applications arealso provided.

BACKGROUND OF THE INVENTION

Classic Control Theory provides a number of techniques for an engineerto use in controller design. Existing controllers for linear, timeinvariant, and single-input single-output plants are categorized in onetaxonomy into three forms: the proportional/integral/derivative (PID)controllers, transfer function based (TFB) controllers, and statefeedback (SF) controllers. The PID controller is defined by theequation:

u=K _(P) e+K _(I) ∫e+K _(D) ė  Equation (1)

where (u) is the control signal and (e) is the error between the setpoint and the process output (y) being controlled. As commonly used, thenotation of a dot (•) above a variable indicates the use of a derivativeof the variable where the order corresponds to the number of dots.

The PID-type of controller has been employed in engineering and otherapplications since the early 1920s. It is an error-based controller thatdoes not require an explicit mathematical model of the plant 110.

Transfer Function Based Controller

The transfer function based (TFB) controller is given in the form of

$\begin{matrix}{{U(s)} = {{{G_{c}(s)}{E(s)}\mspace{31mu} {G_{c}(s)}} = \frac{n(s)}{d(s)}}} & {{Equation}\mspace{14mu} (2)}\end{matrix}$

where U(s) and E(s) are Laplace Transforms of u and e defined above, andn(s) and d(s) are polynomials in s. The TFB controller can be designedusing methods in control theory based on the transfer function model ofthe plant, Gp(s). A PID controller can be considered a special case of aTFB controller because it has an equivalent transfer function of s.

$\begin{matrix}{{G_{c}(s)} = {k_{p} + \frac{k_{i}}{s} + {k_{d}s}}} & {{Equation}\mspace{14mu} (3)}\end{matrix}$

State Feedback (SF) Controller

The State Feedback (SF) controller can be defined by

u=r+Kx  Equation (4)

where u is the control input, r is the setpoint for the output to followand x is the state vector associated with the system, based on the statespace model of the plant 110:

{dot over (x)}(t)=Ax(t)+Bu(t)  Equation (5)

y(t)=Cx(t)+Du(t)  Equation (6)

where x(t) is the state vector, y(t) is the output vector, u(t) is theinput or control vector, A is the state matrix, B is the input matrix, Cis the output matrix and D is the feed-through matrix. When the state xis not accessible, a state observer (SO):

{circumflex over (x)}=A{circumflex over (x)}+Bu+L(y−ŷ)  Equation (7)

is usable to estimate the state x, where the (̂) symbol is used to denotean estimate or observed value of a given variable and L denotes theobserver coefficient matrix.

State Observers

Observers extract real-time information of a plant's 110 internal statefrom its input-output data. The observer usually presumes precise modelinformation of the plant 110, since performance is largely based on itsmathematical accuracy. Most prior art closed loop controllers requireboth types of information. Such presumptions, however, often make themethod impractical in engineering applications, since the challenge forindustry remains in constructing these models as part of the designprocess. Another level of complexity is added when gain scheduling andadaptive techniques are used to deal with nonlinearity and timevariance, respectively.

Disturbance Observers and Disturbance Rejection

Recently, disturbance rejection techniques have been used to account foruncertainties in the real world and successfully control complexnonlinear systems. The premise is to solve the problem of model accuracyin reverse by modeling a system with an equivalent input disturbance dthat represents any difference between the actual plant P and aderived/selected model P_(n) of the plant, including externaldisturbances w. An observer is then designed to estimate the disturbancein real time and provide feedback to cancel it. As a result, theaugmented system acts like the model P_(n) at low frequencies, makingthe system behave like P_(n) and allowing a controller to be designedfor P_(n).

The most common of these techniques is the disturbance observer (DOB)structure (Endo, S., H. Kobayashi, C J. Kempf, S. Kobayashi, M. Tomizukaand Y. Hori (1996). “Robust Digital Tracking Controller Design forHigh-Speed Positioning Systems.” Control Eng Practice, 4:4, 527-536;Kim, B. K., H.-T. Choi, W. K. Chung and I. H. Suh (2002). “Analysis andDesign of Robust Motion Controllers in the Unified Framework.” J. ofDynamic Systems, Measurement, and Control, 124, 313-321; Lee, H. S. andM. Tomizuka (1996). “Robust Motion Controller Design for High-AccuracyPositioning Systems.” IEEE Trans. Ind. Electron., 43:1, 48-55; Tesfaye,A., H. S. Lee and M. Tomizuka (2000). “A Sensitivity OptimizationApproach to Design of a Disturbance Observer in Digital Motion Control.”IEEE/ASME Trans, on Mechatronics, 5:1, 32-38; Umeno, T. and Y. Hori(1991). “Robust Speed Control of DC Servomotors Using Modern Two Degreesof-Freedom Controller Design”. IEEE Trans. Ind. Electron., 38:5,363-368). It uses a disturbance observer consisting of a binomialQ-filters and the inverse model of the plant. And the controller iscalled parameterized in the sense that the Q-filter is treated as aparameter. A model deliberately different from P is suggested in E.Schrijver and J. Van Dijk, “Disturbance Observers for Rigid MechanicalSystems: Equivalence, Stability, and Design,” Journal of DynamicSystems, Measurement, and Control, vol. 124, no. 4, pp. 539-548, 2002 tofacilitate design, but no guidelines are given other than it should beas simple as possible, while cautioning stability and performance may bein danger if the model is significantly different from the plant.Another obstacle is that a separate state observer must be designed toprovide state feedback to the controller. In existing research,derivative approximates are used in this way but their effect onperformance and stability is questionable. Furthermore, the controllerdesign is dependent on the DOB design, meaning that derivativeapproximates can not be arbitrarily selected.

Multiple DOBs were used to control a multivariable robot by treating itas a set of decoupled single-input single-output (SISO) systems, eachwith disturbances that included the coupled dynamics (Bickel, R. and M.Tomizuka (1999). “Passivity-Based Versus Disturbance Observer BasedRobot Control: Equivalence and Stability.” J. of Dynamic Systems,Measurement, and Control, 121, 41-47; Hori, Y., K. Shimura and M.Tomizuka (1992). “Position/Force Control of Multi-Axis Robot ManipulatorBased on the TDOF Robust Servo Controller For Each Joint.” Proc. of ACC,753-757; Kwon, S. J. and W. K. Chung (2002). “Robust Performance of theMultiloop Perturbation Compensator.” IEEE/ASME Trans. Mechatronics, 7:2,190-200; Schrijver, E. and J. Van Dijk (2002) Disturbance Observers forRigid Mechanical Systems: Equivalence, Stability, and Design.” J. ofDynamic Systems, Measurement, and Control, 124, 539-548.)

Another technique, referred to as the unknown input observer (UIO),estimates the states of both the plant and the disturbance by augmentinga linear plant model with a linear disturbance model (Burl, J. B.(1999). Linear Optimal Control, pp. 308-314. Addison Wesley Longman,Inc., California; Franklin, G. F., J. D. Powell and M. Workman (1998).Digital Control of Dynamic Systems, Third Edition, Addison WesleyLongman, California; Johnson, C D. (1971). “Accommodation of ExternalDisturbances in Linear Regulator and Servomechanism Problems.” IEEETrans. Automatic Control, AC-16:6, 635-644; Liu, C-S., and H. Peng(2002). “Inverse-Dynamics Based State and Disturbance Observer forLinear Time-Invariant Systems.” J. of Dynamic Systems, Measurement, andControl, 124, 375-381; Profeta, J. A. III, W. G. Vogt and M. H. Mickle(1990). “Disturbance Estimation and Compensation in Linear Systems.”IEEE Trans. Aerospace and Electronic Systems, 26:2, 225-231; Schrijver,E. and J. Van Dijk (2002) “Disturbance Observers for Rigid MechanicalSystems: Equivalence, Stability, and Design.” J. of Dynamic Systems,Measurement, and Control, 124, 539-548). Unlike the DOB structure, thecontroller and observer can be designed independently, like a Luenbergerobserver. However, it still relies on a good mathematical model and adesign procedure to determine observer gains. An external disturbance wis generally modeled using cascaded integrators (l/s^(h)). When they areassumed to be piece-wise constant, the observer is simply extended byone state and still demonstrates a high degree of performance.

Extended State Observer (ESO)

In this regard, the extended state observer (ESO) is quite different.Originally proposed by Han, J. (1999). “Nonlinear Design Methods forControl Systems.” Proc. 14th IFAC World Congress, in the form of anonlinear UIO and later simplified to a linear version with one tuningparameter by Gao, Z. (2003). “Scaling and Parameterization BasedController Tuning.” Proc. of ACC, 4989-4996, the ESO combines the stateand disturbance estimation power of a UIO with the simplicity of a DOB.One finds a decisive shift in the underlying design concept as well. Thetraditional observer is based on a linear time-invariant model thatoften describes a nonlinear time-varying process. Although the DOB andUIO reject input disturbances for such nominal plants, they leave thequestion of dynamic uncertainty mostly unanswered. The ESO, on the otherhand, addresses both issues in one simple framework by formulating thesimplest possible design model P_(d)=l/s^(n) for a large class ofuncertain systems. The design model P_(d) is selected to simplifycontroller and observer design, forcing P to behave like the designmodel P_(d) at low frequencies rather than P_(n). As a result, theeffects of most plant dynamics and external disturbances areconcentrated into a single unknown quantity. The ESO estimates thisquantity along with derivatives of the output, giving way to thestraightforward design of a high performance, robust, easy to use andaffordable industrial controller.

Active Disturbance Rejection Control (ADRC)

Originally proposed by Han, J. (1999). “Nonlinear Design Methods forControl Systems.” Proc. 14th IFAC World Congress, a nonlinear,non-parameterized active disturbance rejection control (ADRC) is amethod that uses an ESO. A linear version of the ADRC controller and ESOwere parameterized for transparent tuning by Gao, Z. (2003). “Scalingand Parameterization Based Controller Tuning.” Proc. of ACC, 4989-4996.Its practical usefulness is seen in a number of benchmark applicationsalready implemented throughout industry with promising results (Gao, Z.,S. Hu and F. Jiang (2001). “A Novel Motion Control Design Approach Basedon Active Disturbance Rejection.” Proc. of 40th IEEE Conference onDecision and Control; Goforth, F. (2004). “On Motion Control Design andTuning Techniques.” Proc. of ACC; Hu, S. (2001). “On High PerformanceServo Control Solutions for Hard Disk Drive.” Doctoral Dissertation,Department of Electrical and Computer Engineering, Cleveland StateUniversity; Hou, Y., Z. Gao, F. Jiang and B. T. Boulter (2001). “ActiveDisturbance Rejection Control for Web Tension Regulation.” Proc. of 40thIEEE Conf on Decision and Control; Huang, Y., K. Xu and J. Han (2001).“Flight Control Design Using Extended State Observer and NonsmoothFeedback.” Proc. of 40th IEEE Conf on Decision and Control; Sun, B andZ. Gao (2004). “A DSP-Based Active Disturbance Rejection Control Designfor a 1 KW H-Bridge DC-DC Power Converter.” IEEE Trans. on Ind.Electronics; Xia, Y., L. Wu, K. Xu, and J. Han (2004). “ActiveDisturbance Rejection Control for Uncertain Multivariable Systems WithTime-Delay., 2004 Chinese Control Conference. It also was applied to afairly complex multivariable aircraft control problem (Huang, Y., K. Xuand J. Han (2001). “Flight Control Design Using Extended State Observerand Nonsmooth Feedback. Proc. of 40th IEEE Conf on Decision andControl).

What is needed is a control framework for application to systemsthroughout industry that are complex and largely unknown to thepersonnel often responsible for controlling them. In the absence ofrequired expertise, less tuning parameters are needed than currentapproaches, such as multi-loop PID, while maintaining or even improvingperformance and robustness.

INDUSTRIAL APPLICATION

A selection of industrial applications is presented herein to provide abackground on some selected applications for the present extended activedisturbance rejection controller.

Tracking Control

Tracking Control refers to the output of a controlled system meetingdesign requirements when a specified reference trajectory is applied.Oftentimes, it refers to how closely the output y compares to thereference input r at any given point in time. This measurement is knownas the error e=r−y.

Control problems are categorized in two major groups; point-to-pointcontrol and tracking control. Point-to-point applications usually callfor a smooth step response with small overshoot and zero steady stateerror, such as when controlling linear motion from one position to thenext and then stopping. Since the importance is placed on destinationaccuracy and not on the trajectory between points, conventional designmethods produce a bandwidth limited controller with inherent phase lagin order to produce a smooth output. Tracking applications require agood tracking of a reference input by keeping the error as small aspossible at all times, not just at the destination. This is particularlycommon in controlling a process that never stops. Since the importanceis placed on accurately following a changing reference trajectorybetween points, phase lags are avoided as much as possible because theymay lead to significant errors in the transient response, which lastsfor the duration of the process in many applications. A common method ofreducing phase lag is to increase the bandwidth, at the cost ofincreased control effort (energy) and decreased stability margin.

Various methods have been used to remove phase lag from conventionalcontrol systems. All of them essentially modify the control law tocreate a desired closed loop transfer function equal to one. As aresult, the output tracks the reference input without much phase lag andthe effective bandwidth of the overall system is improved. The mostcommon ethod is model inversion where the inverse of the desired closedloop transfer function is added as a prefilter. Another method proposeda Zero Phase Error Tracking Controller (ZPETC) that cancels poles andstable zeros of the closed loop system and compensates for phase errorintroduced by un-cancelable zeros. Although it is referred to as atracking controller, it is really a prefilter that reduces to theinverse of the desired closed loop transfer function when unstable zerosare not present. Other methods consist of a single tracking control lawwith feed forward terms in place of the conventional feedback controllerand prefilter, but they are application specific. However, all of theseand other previous methods apply to systems where the model is known.

Model inaccuracy can also create tracking problems. The performance ofmodel-based controllers is largely dependent on the accuracy of themodel. When linear time-invariant (LTI) models are used to characterizenonlinear time-varying (NTV) systems, the information becomes inaccurateover time. As a result, gain scheduling and adaptive techniques aredeveloped to deal with nonlinearity and time variance, respectively.However, the complexity added to the design process leads to animpractical solution for industry because of the time and level ofexpertise involved in constructing accurate mathematical models anddesigning, tuning, and maintaining each control system.

There have been a number of high performance tracking algorithms thatconsist of three primary components: disturbance rejection, feedbackcontrol, and phase lag compensation implemented as a prefilter. First,disturbance rejection techniques are applied to eliminate modelinaccuracy with an inner feedback loop. Next, a stabilizing controlleris constructed based on a nominal model and implemented in an outerfeedback loop. Finally, the inverse of the desired closed loop transferfunction is added as a prefilter to eliminate phase lag. Many studieshave concentrated on unifying the disturbance rejection and controlpart, but not on combining the control and phase lag compensation part,such as the RIC framework. Internal model control (IMC) cancels anequivalent output disturbance. B. Francis and W. Wonham, “The InternalModel Principal of Control Theory,” Automatica, vol 12, 1976, pp.457-465. E. Schrijver and J. Van Dijk, “Disturbance Observers for RigidMechanical Systems: Equivalence, Stability, and Design,” Journal ofDynamic Systems, Measurement, and Control, vol. 124, December 2002, pp.539-548 uses a basic tracking controller with a DOB to control amultivariable robot. The ZPETC has been widely used in combination withthe DOB framework and model based controllers.

Thus, having reviewed prior art in tracking control, the application nowdescribes example systems and methods of tracking control employingpredictive ADRC.

Web Processing Applications

Web tension regulation is a challenging industrial control problem. Manytypes of material, such as paper, plastic film, cloth fabrics, and evenstrip steel are manufactured or processed in a web form. The quality ofthe end product is often greatly affected by the web tension, making ita crucial variable for feedback control design, together with thevelocities at the various stages in the manufacturing process. Theever-increasing demands on the quality and efficiency in industrymotivate researchers and engineers alike to explore better methods fortension and velocity control. However, the highly nonlinear nature ofthe web handling process and changes in operating conditions(temperature, humidity, machine wear, and variations in raw materials)make the control problem challenging.

Accumulators in web processing lines are important elements in webhandling machines as they are primarily responsible for continuousoperation of web processing lines. For this reason, the study andcontrol of accumulator dynamics is an important concern that involves aparticular class of problems. The characteristics of an accumulator andits operation as well as the dynamic behavior and control of theaccumulator carriage, web spans, and tension are known in the art.

Both open-loop and closed-loop methods are commonly used in webprocessing industries for tension control purposes. In the open-loopcontrol case, the tension in a web span is controlled indirectly byregulating the velocities of the rollers at either end of the web span.An inherent drawback of this method is its dependency on an accuratemathematical model between the velocities and tension, which is highlynonlinear and highly sensitive to velocity variations. Nevertheless,simplicity of the controller outweighs this drawback in manyapplications. Closing the tension loop with tension feedback is anobvious solution to improve accuracy and to reduce sensitivity tomodeling errors. It requires tension measurement, for example, through aload cell, but is typically justified by the resulting improvements intension regulation.

Most control systems will unavoidably encounter disturbances, bothinternal and external, and such disturbances have been the obstacles tothe development of high performance controller. This is particularlytrue for tension control applications and, therefore, a good tensionregulation scheme must be able to deal with unknown disturbances. Inparticular, tension dynamics are highly nonlinear and sensitive tovelocity variations. Further, process control variables are highlydependent on the operating conditions and web material characteristics.Thus, what are needed are systems and methods for control that are notonly overly dependent on the accuracy of the plant model, but alsosuitable for the rejection of significant internal and externaldisturbances.

Jet Engine Control Applications

A great deal of research has been conducted towards the application ofmodern multivariable control techniques on aircraft engines. Themajority of this research has been to control the engine at a singleoperating point. Among these methods are a multivariable integratorwindup protection scheme (Watts, S. R. and S. Garg (1996). “An OptimizedIntegrator Windup Protection Technique Applied to a Turbofan EngineControl,” AIAA Guidance Navigation and Control Conf.), a tracking filterand a control mode selection for model based control (Adibhatla S. andZ. Gastineau (1994). “Tracking Filter Selection And Control ModeSelection For Model Based Control.” AIAA 30th Joint PropulsionConference and Exhibit.), an H, method and linear quadratic Gaussianwith loop transfer recovery method (Watts, S. R. and S. Garg (1995).” AComparison Of Multivariable Control Design Techniques For A TurbofanEngine Control.” International Gas Turbine and Aeroengine Congress andExpo.), and a performance seeking control method (Adibhatla, S. and K.L. Johnson (1993). “Evaluation of a Nonlinear Psc Algorithm on aVariable Cycle Engine.” AIAA/SAE/ASME/ASEE 29th Joint PropulsionConference and Exhibit.). Various schemes have been developed to reducegain scheduling (Garg, S. (1997). “A Simplified Scheme for SchedulingMultivariable Controllers.” IEEE Control Systems) and have even beencombined with integrator windup protection and H_(m) (Frederick, D. K.,S. Garg and S. Adibhatla (2000). “Turbofan Engine Control Design UsingRobust Multivariable Control Technologies. IEEE Trans. on ControlSystems Technology).

Conventionally, there have been a limited number of control techniquesfor full flight operation (Garg, S. (1997). “A Simplified Scheme forScheduling Multivariable Controllers.” IEEE Control Systems; and Polley,J. A., S. Adibhatla and P J. Hoffman (1988). “Multivariable TurbofanEngine Control for Full Conference on Decision and Control FlightOperation.” Gas Turbine and Expo). However, there has been nodevelopment of tuning a controller for satisfactory performance whenapplied to an engine. Generally, at any given operating point, modelscan become inaccurate from one engine to another. This inaccuracyincreases with model complexity, and subsequently design and tuningcomplexity. As a result, very few of these or similar aircraft designstudies have led to implementation on an operational vehicle.

The current method for controlling high performance jet engines remainsmultivariable proportional-integral (PI) control (Edmunds, J. M. (1979).“Control System Design Using Closed-Loop Nyquist and Bode Arrays.” Int.J. on Control, 30:5, 773-802, and Polley, J. A., S. Adibhatla and P J.Hoffman (1988). “Multivariable Turbofan Engine Control for FullConference on Decision and Control. Flight Operation.”Gas Turbine andExpo). Although the controller is designed by implementing Bode andNyquist techniques and is tunable, a problem remains due to the sheernumber of tuning parameters compounded by scheduling.

SUMMARY OF INVENTION

The present system, method, process, and device for controlling systemsis applicable for a variety of different application and has utility ina number of different industrial fields. The following brief summarysection is provided to facilitate a basic understanding of the natureand capabilities of the present system and method. This summary sectionis not an extensive nor comprehensive overview and is not intended toidentify key or critical elements of the present systems or methods orto delineate the scope of these items. Rather this brief summary isintended to provide a conceptual introduction in simplified form as anintroduction for the more detailed description presented later in thisdocument.

The present application describes selected embodiments of the ADRCcontroller that comprise one or more computer components, that extend,build upon and enhance the use of ADRC controllers and provide enhancedperformance and utility. Specifically, in one aspect the ADRC controllerutilizes a predictive computer component or module that in oneembodiment predicts future values of the plant output and in a secondembodiment predicts future estimates of the system state and generalizeddisturbance. The generalized disturbance includes dynamics of the plantitself such that the plant is effectively reduced to a cascaded integralplant. In the various embodiments a variety of estimates are availableand introduced.

In still another aspect, the ADRC controller further includes a model ofthe plant dynamics. The model is used in one embodiment to improve thestate estimator or predictor. In a second embodiment the model is usedto provide an enhanced control law that provides improved performance.In both aspects of the embodiments discussed in this paragraph, thegeneralized disturbance captures model errors and discrepancies andallows the system to eliminate errors caused by model errors,discrepancies, and assumptions. In the multiple embodiments, a varietyof models are available for use by one of ordinary skill in the art toachieve the desired performance relative to the type of plant beingcontrolled.

In yet another aspect, the ADRC controller further includes anon-linear, or discrete time optimal control law. The non-linear controllaw of the present embodiment utilizes non-linear control laws toimprove the overall performance of an ADRC controller as compared to anADRC controller with a linear proportional-derivative based control law.

In still another aspect, additional knowledge of derivative plant outputgathered from high quality sensor information or direct measurementsensors (e.g. velocity and acceleration sensors) is used to reduce theorder of the system state and disturbance estimator.

BRIEF DESCRIPTION OF THE DRAWINGS

The accompanying figures depict multiple embodiments of a device andmethod for the separation of constituents from a flow. A briefdescription of each figure is provided below. Elements with the samereference numbers in each figure indicate identical or functionallysimilar elements. Additionally, the left-most digit(s) of a referencenumber identifies the drawings in which the reference number firstappears.

FIG. 1 is a block diagram of a Prior Art feedback controller.

FIG. 2 is a block diagram of an ADRC configuration for a 2nd orderplant.

FIG. 3 is a block diagram of a predictive ADRC configuration.

FIG. 4 is a block diagram of a 2 degree of freedom transfer functionbased (TFB) control structure.

FIG. 5 is a block diagram of an implementation of a PADRC configurationbased on output prediction.

The present invention is described with reference to block diagrams andoperational flow charts. It is to be understood that the functions/actsnoted in the blocks may occur out of the order noted in the operationalillustrations. For example, two blocks shown in succession may in factbe executed substantially concurrently or the blocks may sometimes beexecuted in the reverse order, depending upon the functionality/actsinvolved, including for example executing as asynchronous threads on aprocessor. Although some of the diagrams include arrows on communicationpaths to show a primary direction of communication, it is to beunderstood that communication may occur in the opposite direction to thedepicted arrows.

DETAILED DESCRIPTION

Multiple embodiments of a system, device, and method for the control ofsystems are presented herein. Those of ordinary skill in the art canreadily use this disclosure to create alternative embodiments using theteaching contained herein.

Lexicon

The following teens used herein have the meanings as follows.

As used herein the term “computer component” refers to a computer andelements of a computer, such as hardware, firmware, software, acombination thereof, or software in execution. For example, a computercomponent can include by way of example, a process running on aprocessor, a processor, an object, an executable, an execution thread, aprogram, and a computer itself. One or more computer components can invarious embodiments reside on a server and the server can be comprisedof multiple computer components. One or more computer components are insome cases referred to as computer systems whereby one or more computercomponents operate together to achieve some functionality. One or morecomputer components can reside within a process and/or thread ofexecution and a computer component can be localized on one computerand/or distributed between two or more computers.

“Software”, as used herein, includes but is not limited to, one or morecomputer readable and/or executable instructions that cause a computeror other electronic device to perform functions, actions and/or behavein a desired manner. The instructions may be embodied in various formslike routines, algorithms, modules, methods, threads, and/or programs.Software may also be implemented in a variety of executable and/orloadable forms including, but not limited to, a stand-alone program, afunction call (local and/or remote), a servelet, an applet, instructionsstored in a memory, part of an operating system or browser, and thelike. It is to be appreciated that the computer readable and/orexecutable instructions can be located in one computer component and/ordistributed between two or more communicating, co-operating, and/orparallel processing computer components and thus can be loaded and/orexecuted in serial, parallel, massively parallel and other manners. Itwill be appreciated by one of ordinary skill in the art that the form ofsoftware may be dependent on, for example, requirements of a desiredapplication, the environment in which it runs, and/or the desires of adesigner/programmer or the like.

“Computer communications”, as used herein, refers to a communicationbetween two or more computers and can be, for example, a networktransfer, a file transfer, an applet transfer, an email, a hypertexttransfer protocol (HTTP) message, a datagram, an object transfer, abinary large object (BLOB) transfer, and so on. A computer communicationcan occur across, for example, a wireless system (e.g., IEEE 802.11), anEthernet system (e.g., IEEE 802.3), a token ring system (e.g., IEEE802.5), a local area network (LAN), a wide area network (WAN), apoint-to-point system, a circuit switching system, a packet switchingsystem, and so on.

An “operable connection” is one in which signals and/or actualcommunication flow and/or logical communication flow may be sent and/orreceived. Usually, an operable connection includes a physical interface,an electrical interface, and/or a data interface, but it is to be notedthat an operable connection may consist of differing combinations ofthese or other types of connections sufficient to allow operablecontrol.

As used herein, the term “signal” may take the form of a continuouswaveform and/or discrete value(s), such as digital value(s) in a memoryor register, present in electrical, optical or other form.

The term “controller” as used herein indicates a method, process, orcomputer component adapted to control a plant (i.e. the system to becontrolled) to achieve certain desired goals and objectives.

As used herein the following symbols are used to describe specificcharacteristics of the variable. A variable with a “̂” overtop thevariable label, unless other indicated to the contrary, indicates thatthe variable is an estimate of the actual variable of the same label. Avariable with a “•” overtop the variable label, unless otherwiseindicated, indicates that the variable is an nth derivative of thatvariable, where n equals the number of dots.

To the extent that the term “includes” is employed in the detaileddescription or the claims, it is intended to be inclusive in a mannersimilar to the term “comprising” as that term is interpreted whenemployed as. a transitional word in a claim.

To the extent that the term “or” is employed in the claims (e.g., A orB) it is intended to mean “A or B or both”. When the author intends toindicate “only A or B but not both”, then the author will employ theterm “A or B but not both”. Thus, use of the term “or” in the claims isthe inclusive, and not the exclusive, use.

Active Disturbance Rejection Control (ADRC)

Active Disturbance Rejection Control (“ADRC”) is a method or processand, when embedded within a computer system, a device (i.e., acontroller that processes information and signals using the ADRCalgorithms), that effectively rejects both unknown internal dynamics ofa system of an arbitrary order to be controlled, or plant 110, andexternal disturbances in real time without requiring detailed knowledgeof the plant 110 dynamics, which is required by most existing controldesign methods.

The only information needed to configure an ADRC controller is knowledgeof the relative order of the plant 110 and the high frequency gain ofthe plant 110. In comparison, traditional model-based control methodsrequire accurate mathematical models to conform with prevailingmodel-based design methods. In some embodiments, the ADRC controllerleverages knowledge gained from model-based controller designs toimprove control performance. The ADRC approach is unique in that itfirst forces an otherwise nonlinear, time-varying and uncertain plant110 to behave as a simple linear cascade integral plant that can beeasily controlled. To this end, the unknown, nonlinear, and time-varyinginternal dynamics is combined with the external disturbances to foiniwhat is denoted as the generalized disturbance, which is then estimatedand rejected, thus greatly simplifying the control design.

The embodiments of the ADRC controller design are founded on the basictheory that uncertainties in dynamics and unknown external disturbancesin the operating environment are a key challenge to be dealt with incontrol engineering practice. Traditional control systems are incapableof providing a solution to controlling and mitigating the impact ofthese uncertainties. The ADRC controller in contrast is specificallyadapted to mitigate and eliminate the impact of these uncertainties,thus providing a dynamically simplified system for control.

One previous major characteristic of the ADRC controller is theparameterization of the controller as a function of control loopbandwidth. This parameterization makes all of the ADRC controllerparameters functions of the control loop bandwidth; see, U.S. patentapplication Ser. No. 10/351,664, titled, “Scaling and Parameterizing AController” to Zhiqiang Gao, which is hereby incorporated by referencein its entirety.

An embodiment of a parameterized ADRC controller applied to asecond-order plant 202 is illustrated in FIG. 2. The system to becontrolled is a second-order plant 202 and represents an arbitrary plant110 with predominantly second-order dynamics. The second-order plant 202has an output (y) in response to an input that comprises a controlsignal (u) with superimposed external disturbances (d).

Consider a general uncertain, nonlinear and time-varying second-orderplant 202 of the form:

ÿ=f(y,{dot over (y)},d,t)+bu  Equation (8)

where y is system output, u is the control signal, b is a constant, d isexternal input disturbance, f(y,{dot over (y)},d,t) is treated as thegeneralized disturbance. When this generalized disturbance is cancelled,the system is reduced to a simple double-integral plant with a scalingfactor b, which can be easily controlled. Here we assume that theapproximate value of b, denoted as {circumflex over (b)}, is given orestimated. As shown in FIG. 2, the scaling factor ({circumflex over(b)}) is used to scale 208, the control output (uft). In otherembodiments, the scaling factor is implicit within the linear PD controllaw 206.

To capture the information of the generalized disturbance f(y,{dot over(y)},d,t) and cancel it from the system dynamics leads us to theExtended State Observer (ESO) 204, which is now presented. Let x₁=y,x₂={dot over (y)}, and x₃=f, the description of the above plant 202 isrewritten in state space form with f(y, {dot over (y)}, d,t), or simplyf, treated as an additional state:

$\begin{matrix}\left\{ {{{\begin{matrix}{\overset{.}{x} = {{A_{x}x} + {B_{x}u} + {E\overset{.}{f}}}} \\{y = {C_{x}x}}\end{matrix}{where}A_{x}} = \begin{bmatrix}0 & 1 & 0 \\0 & 0 & 1 \\0 & 0 & 0\end{bmatrix}},{B_{x} = \begin{bmatrix}0 \\b \\0\end{bmatrix}},{E = \begin{bmatrix}0 \\0 \\1\end{bmatrix}},{C_{x} = \left\lbrack {1\mspace{14mu} 0\mspace{14mu} 0} \right\rbrack}} \right. & {{Equation}\mspace{14mu} (9)}\end{matrix}$

The corresponding state observer is

$\begin{matrix}\left\{ \begin{matrix}{\overset{.}{z} = {{A_{x}z} + {B_{z}u} + {L\left( {y_{m} - {C_{x}z}} \right)}}} \\{\hat{f} = {Vz}}\end{matrix} \right. & {{Equation}\mspace{14mu} (10)}\end{matrix}$

where y_(m) is the measured system output, B_(z)=[0 {circumflex over(b)} 0]^(T), V=[0 0 1], and L=[l₁ l₂ l₃]^(T) is the observer gainvector. With the parameterization technique as described more fully inU.S. patent application Ser. No. 10/351,664, L is obtained by solvingthe equation λ(s)=|sI−A_(x)+Lc_(x)|=(s+ω_(o))³; that is, all theeigenvalues of the observer are placed at −ω_(o) with

l ₁=3ω_(o) ,l ₂=3ω_(o) ² ,l ₃=ω_(o) ³  Equation (11)

With a well-tuned extended state observer 204, z₁, z₂, and z₃ willclosely track y, {dot over (y)}, and f respectively, i.e. z₁≈y, z₂≈{dotover (y)}, and; z_(3≈f)

Using the information provided by the extended state observer 204, thegeneralized disturbance f is rejected by the control law

u=(u ₀ −z ₃)/{circumflex over (b)}  Equation (12)

which reduces the plant 110 to a unity gain, double integral plant

ÿ=u _(o)  Equation (13)

that can be easily controlled with linear proportional-derivative (PD)with feedforward

u _(o) =g _(pd)(e,ë,ω _(c))+{umlaut over (r)}  Equation (14)

where g_(pd) (e, ė, ω_(c)) is the parameterized linear PD controllerwith the loop bandwidth of ω_(c)

g _(pd)(e,ė,ω _(c))=ω_(c) ² e+2ω_(c) ė  Equation (15)

with the error term defined as

e=r−z ₁ ė={dot over (r)}−z ₂  Equation (16)

where the desired trajectory (r) is provided by the user.

Note that the linear PD control law 206 in Equation (14) corresponds tothe common PD gains of

k _(p)=ω_(c) ² and k _(d)=2ω_(c)  Equation (17)

Using these details, the ADRC controller is generalized in its variousembodiments to control an arbitrary n^(th)-order system y^((n))=f(y,{dotover (y)}, . . . ,y^((n−1)),d,t)+bu, in which the external disturbancesand unknown internal dynamics are rejected. In a similar manner, theplant 110 is reduced to a cascaded integral plant y^((n))=u, which isthen controlled by a generalized proportional-derivative (PD) controllaw, also described as a linear PD control law 206 similar to Equation(14).

Use of Enhanced Sensor Information to Improve Control Performance

In other embodiments of the ADRC controller, the linear PD control law206 is updated to enhance performance by using direct measurement of thesystem output states (y_(m)) instead of the estimated states (z₁, z₂).Specifically, when the measured output y, (t) is sufficiently noise freeand {dot over (y)}_(m)(t) is obtainable, either via direct measurementor from y_(m)(t), the control law in Equation (14) can be implementedwith

e=r−y _(m) ė={dot over (r)}−{dot over (y)} _(m)  Equation (18)

That is, the output measurement and its derivative are used in thecontrol law, instead of their estimates, z₁ and z₂. Similarly for then^(th)-order plant 110, the output measurement y_(m) and its derivativesfrom the 1^(st) to the (n−1)^(th)-order available, can also be useddirectly in the PD control law, also referred to as the linear PDcontrol law 206, instead of their estimates from the Extended StateObserver (ESO) 204. Using the direct measurement of the outputmeasurement state y_(m) and its derivatives reduces the phase lag andimprove control system performance, if the output measurement and itsderivatives are not corrupted by noise.

Multiple Extended States

The Extended State Observer 204 in alternative embodiments includes morethan one extended states, such as embodiments where the 2nd order plant202 is replaced with a higher order plant 110. In one embodiment, whenaugmented with two extended states, x₃=f and x₄={dot over (f)}, theextended plant state space model can be written as follows.

$\begin{matrix}\left\{ {{{\begin{matrix}{\overset{.}{x} = {{A_{x}x} + {B_{x}u} + {E\overset{¨}{f}}}} \\{y = {C_{x}x}}\end{matrix}{where}A_{x}} = \begin{bmatrix}0 & 1 & 0 & 0 \\0 & 0 & 1 & 0 \\0 & 0 & 0 & 1 \\0 & 0 & 0 & 0\end{bmatrix}},{B_{x} = \begin{bmatrix}0 \\b \\\begin{matrix}0 \\0\end{matrix}\end{bmatrix}},{E = \begin{bmatrix}0 \\0 \\0 \\1\end{bmatrix}},{C_{x} = \left\lbrack {1\mspace{14mu} 0\mspace{14mu} 0\mspace{20mu} 0} \right\rbrack}} \right. & {{Equation}\mspace{14mu} (19)}\end{matrix}$

With z₁, z₂, z₃, and z₄ tracking x₁, x₂, x₃, and x₄ respectively, the4^(th)-order ESO 204 is employed to estimate f.

$\begin{matrix}\left\{ \begin{matrix}{\overset{.}{z} = {{A_{x}z} + {B_{z}u} + {L\left( {y_{m} - {C_{x}z}} \right)}}} \\{\hat{f} = {Vz}}\end{matrix} \right. & {{Equation}\mspace{14mu} (20)}\end{matrix}$

where B_(z)=[0 {circumflex over (b)} 0 0]^(T) and V=[0 0 1 0].

In this manner, any embodiment of an ADRC controller is adaptable foruse with an arbitrary number (h) extended state observer states with isgeneralized to control an n^(th)-order system.

Discrete Implementation

ADRC algorithms are implemented in discrete time, with ESO 204discretized in both Euler and ZOH, and in both the current estimator andpredictive estimator form.

In discrete implementation, the ESO 204 pole location is determined inthe Z domain by solving the equation λ(z)=|zI−φ+L_(p)C_(x)|=(Z−β)³,where β=e^(−ω) ^(o) ^(T).

Predictive Discrete Extended State Observer (PDESO) is in predictiveestimator form, as shown in the following equation.

$\begin{matrix}\left\{ \begin{matrix}{{z\left\lbrack {k + 1} \right\rbrack} = {{\Phi \; {z\lbrack k\rbrack}} + {\Gamma \; {u\lbrack k\rbrack}} + {L_{p}\left( {{y_{m}\lbrack k\rbrack} - {z_{1}\lbrack k\rbrack}} \right)}}} \\{{\overset{\_}{z}\lbrack k\rbrack} = {z\lbrack k\rbrack}}\end{matrix} \right. & {{Equation}\mspace{14mu} (21)}\end{matrix}$

Current Discrete Extended State Observer (CDESO) is in current estimatorform, as shown in the following equation.

$\begin{matrix}\left\{ \begin{matrix}{{z\left\lbrack {k + 1} \right\rbrack} = {{\Phi \; {z\lbrack k\rbrack}} + {\Gamma \; {u\lbrack k\rbrack}} + {L_{p}\left( {{y_{m}\lbrack k\rbrack} - {z_{1}\lbrack k\rbrack}} \right)}}} \\{{\overset{\_}{z}\lbrack k\rbrack} = {{z\lbrack k\rbrack} + {L_{c}\left( {{y_{m}\lbrack k\rbrack} - {z_{1}\lbrack k\rbrack}} \right)}}}\end{matrix} \right. & {{Equation}\mspace{14mu} (22)}\end{matrix}$

The gain matrices of different ADRC discrete implementations are shownin the following Table I.

TABLE I PDESO AND CDESO COEFFICIENT MATRICES Type Φ Γ L_(p) L_(c) Euler$\quad\begin{bmatrix}1 & T & 0 \\0 & 1 & T \\0 & 0 & 1\end{bmatrix}$ $\quad\begin{bmatrix}0 \\{\hat{b}T} \\0\end{bmatrix}$ $\quad\begin{bmatrix}{3\left( {1 - \beta} \right)} \\{3{\left( {1 - \beta} \right)^{2}/T}} \\{\left( {1 - \beta} \right)^{3}/T^{2}}\end{bmatrix}$ $\quad\begin{bmatrix}{1 - \beta^{3}} \\{\left( {1 - \beta} \right)^{2}{\left( {2 + \beta} \right)/T}} \\{\left( {1 - \beta} \right)^{3}/T^{2}}\end{bmatrix}$ ZOH $\quad\begin{bmatrix}1 & T & \frac{T^{2}}{2} \\0 & 1 & T \\0 & 0 & 1\end{bmatrix}$ $\quad\begin{bmatrix}{\hat{b}\frac{T^{2}}{2}} \\{\hat{b}T} \\0\end{bmatrix}$ $\quad\begin{bmatrix}{3\left( {1 - \beta} \right)} \\{3\left( {1 - \beta} \right)^{2}{\left( {5 + \beta} \right)/2}T} \\{\left( {1 - \beta} \right)^{3}/T^{2}}\end{bmatrix}$ $\quad\begin{bmatrix}{1 - \beta^{3}} \\{3\left( {1 - \beta} \right)^{2}{\left( {1 + \beta} \right)/2}T} \\{\left( {1 - \beta} \right)^{3}/T^{2}}\end{bmatrix}$

This discrete implementation of ADRC is also generalizable ton^(th)-order plants 110.

Predictive Active Disturbance Rejection Controller

The thrust of the many of the embodiments presented here is the additionof the predictive functions to extend the capabilities of the originalcomponents of the ADRC. This extends the ADRC controller and generatesembodiments of predictive versions of the ADRC, such as the predictiveADRC 300, depicted in block diagram form in FIG. 3 In this embodiment,the linear PD control law 206 in the predictive ADRC 300 is now apredictive control law 306. Similarly, the observer, or morespecifically the extended state observer 204 from the traditional ADRC200 is now embodied as a predictive state and disturbance observer 304.In one embodiment the predictive state and disturbance observer 304 is apredictive extended state observer (PESO) where the prediction occurswithin the bounds of the ESO 204 to where the estimated states z_(n)(t)are predicted to form the predictive state and disturbance observer 304,and outputs the prediction of the estimated states z_(n)(t+τ). In asecond embodiment, a state predictor method is used to extend the designof the ESO 204 to estimate the states z_(n)(t) and form the predictivestate and disturbance observer 304.

In other embodiments, other estimators and predictors, either in anintegrated form or in combination are used to generate similarinformation as that generated by the predictive state and disturbanceobserver 304 and used by the embodiments predictive ADRC 300. One ofthese other embodiments adopts a disturbance observer (DOB) estimatorstructure to estimate the disturbance and a state observer and apredictor element that together form the predictive state anddisturbance observer 304. In a second other embodiment, the predictivestate and disturbance observer 304 utilizes an unknown input observer(UIO) estimator. Other estimators, including combinations of estimatorssuch as the combination of a state observer and a DOB, are suitable forestimation of both the plant 110 output and the known part of thegeneral disturbance (f_(n)) are adaptable using the disclosure providedherein to those of ordinary skill in the art to serve as a predictivestate and disturbance observer 304.

The challenge in constructing the various embodiments of a predictivecontroller within the general purpose ADRC controller architecture ishow the various signals used by the ADRC algorithm are predicted andused. The signals used by the ADRC algorithm originate from a multitudeof sources in the various embodiments, including in one embodiment theextended state observer 204 (ESO). In another embodiment, a simplified,reduced-order ESO (RESO) is used when either: (i) quality measurementsof the output signals allows numerical differentiation to estimatederivatives of the output signal; or (ii) direct measurement ofderivatives are possible (e.g. velocity and acceleration sensors), areavailable.

In another embodiment, a nonlinear proportional-derivative (NPD) controllaw is used to further optimize the performance of the present ADRCsystem. The RESO aspect of the ADRC controller provides alternativeembodiments of ESO 204 and RESO; a general form of RESO with multipleextended states for an n^(th)-order plant 110, and incorporation of theknown plant 110 dynamics into ESO 204 and control law.

Furthermore, to facilitate practical implementations of the newtechniques, we have provided at selected locations herein, the transferfunction equivalent predictive ADRC 400, with H(s) 402 and G_(c)(s) 404,in a two-degree-of-freedom (2dof) structure shown in FIG. 4. Similarly,for a transfer function equivalent predictive ADRC 400, the plantdynamics are represented as G_(p)(s) 402. For various forms of thePredictive ADRC 400 described herein, these transfer functionexpressions, together with the NPD and RESO, are novel and significanteven when the amount of prediction is set to zero.

All of these new developments' help to make the predictive ADRC 300possible and to provide the means of implementation the presentembodiments in a diverse range of industrial machinery, and they areillustrated in details in the following sections.

Basic Design Principles for Predictive ADRC

In the real world, the sensor feedback, the generation of the controlsignal and its arrival at the actuator, and the estimation of the statesand disturbances all contain time delays and phase lags, which canendanger system stability. To accommodate such delays and lags, wepropose the following Predictive ADRC control law:

u(t)=(g(e(t+l _(e)),{dot over (e)}(t+l _(ė)),ω_(c))+{umlaut over(r)}(t+l _(u))−{circumflex over (f)}(t+l _(f)))/{circumflex over(b)}  Equation (23)

where g(e,ė,ω_(c)) is the parameterized linear PD controller 206, ornonlinear proportional derivative (NPD) controller, Equation (68),l_(e), l_(ė), l_(u), and l_(f) are positive real numbers representingthe prediction horizons, a special case of which isl_(e)=l_(ė)=l_(u)=l_(f)=τ and the corresponding control law is

u(t)=(g(e(t+τ),{dot over (e)}(t+τ),ω_(c))+{umlaut over(r)}(t+τ)−{circumflex over (f)}(t+τ))/{circumflex over (b)}  Equation(24)

Also, if a part of the plant 110 dynamics, f_(n), is given, it can beincorporated into ADRC, as shown later in this disclosure. In thecontext of Predictive ADRC 300, the knowledge of f_(n) can be used tohelp predict its future value, f_(n)(t+l_(f) _(n) ). The correspondingcontrol law for Equation (23) and Equation (24) are

u(t)=(g(e(t+l _(e)),{dot over (e)}(t+l _(ė)),ω_(c))+{umlaut over(r)}(t+l _(u))−f _(n)(t+l _(f) _(n) )−{circumflex over (f)} _(u)(t+l_(f) _(u) ))/{circumflex over (b)}  Equation (25)

and

u(t)=(g(e(t+τ),{dot over (e)}(t+τ),ω_(c))+{umlaut over (r)}(t+τ)−f_(n)(t+τ)−{circumflex over (f)} _(u)(t+τ))/{circumflex over(b)}  Equation (26)

respectively.

Described below are various methods to obtain the prediction of e(t),ė(t), and {circumflex over (f)}(t), respectively, so that thesepredictive ADRC 300 control laws can be implemented.

Prediction of e and ė

Two embodiments of methods for predicting the values of error e(t) andits derivative are presented herein.

Method 1:

Using Taylor Series, the prediction of e(t), ė(t) can be obtained ase(t+10=e(t)+e(t)l_(e) or

${{e\left( {t + l_{e}} \right)} = {{e(t)} + {{\overset{.}{e}(t)}l_{e}} + {\frac{\overset{¨}{e}(t)}{2}l_{e}^{2}}}},$

and ė(t+l_(ė))=ė(t)+ë(t)l_(ė). Here ė(t) can be obtained directly from{dot over (r)}−{dot over (y)}, or from {dot over (r)}−z₂ using regularESO, or {dot over (r)}−z₁ using RESO. Here ë(t) can be obtained asë(t)={umlaut over (r)}(t)−ÿ(t)≈{umlaut over (r)}(t)−(z₃(t)+{circumflexover (b)}u(t)) with ESO or ë(t)={umlaut over (r)}(t)−ÿ(t)≈{umlaut over(r)}(t)−(z₂(t)+{circumflex over (b)}u(t)) with RESO.

Method 2:

Since e(t+τ)=r(t+τ)−z₁(t+τ) and ė(t+τ)={dot over (r)}(t+τ)−z₂(t+τ),e(t+τ) and ė(t+τ) can be obtained directly using the future referencesignals r(t+τ) and {dot over (r)}(t+τ), which are generally known, andthe predicted output of the ESO z₁(t+τ) and z₂ (t+τ), which need to beobtained approximately.

Prediction of {circumflex over (f)} in ESO and RESO

The primary method of predicting {circumflex over (f)} is the 1^(st) or2^(nd)-order Taylor Series approximation, i.e.

$\begin{matrix}{{{\hat{f}\left( {t + l_{f}} \right)} \approx {{\hat{f}(t)} + {{\hat{\overset{.}{f}}(t)}l_{f}}}},{or}} & {{Equation}\mspace{14mu} (27)} \\{{\hat{f}\left( {t + l_{f}} \right)} \approx {{\hat{f}(t)} + {{\hat{\overset{.}{f}}(t)}l_{f}} + {\frac{\hat{\overset{¨}{f}}(t)}{2}l_{f}^{2}}}} & {{Equation}\mspace{14mu} (28)}\end{matrix}$

which requires the 1^(st) and/or 2^(nd) order derivatives of {circumflexover (f)}. To this end, a new form and a new way of implementation ofextended state observer (ESO) 204 are introduced first, followed by thediscussion on how to use them to obtain the predicted estimate of{circumflex over (f)} in Equation (27) and Equation (28)

Reduced-Order ESO

In the observer-based control methods, the phase lag introduced by theobserver decreases the phase margin of the control loop and is thereforeundesired. To reduce the phase lag in the ESO 204, an embodiment of areduced-order ESO (RESO) described here.

Consider the plant 202 in Equation (9), if the derivative of themeasured output y_(m) ({dot over (y)}_(m)), is given, a RESO can beconstructed with z₁ and z₂ estimating x₂={dot over (y)} and x₃=frespectively. The correspondence between the augmented plant state spacemodel and the RESO is shown as follows.

$\begin{matrix}{\mspace{76mu} {{\overset{.}{x} = {{\begin{bmatrix}0 & 1 & 0 \\0 & 0 & 1 \\0 & 0 & 0\end{bmatrix}\mspace{14mu} X} + {\begin{bmatrix}0 \\b \\0\end{bmatrix}u} + {\begin{bmatrix}0 \\0 \\1\end{bmatrix}\overset{.}{f}}}}\mspace{79mu} {y = {{\left\lbrack {1\mspace{14mu} 0\mspace{14mu} 0} \right\rbrack {x\; \begin{bmatrix}{\overset{.}{z}}_{1} \\{\overset{.}{z}}_{2}\end{bmatrix}}} = {{\begin{bmatrix}0 & 1 \\0 & 0\end{bmatrix}\left\lbrack \begin{matrix}z_{1} \\z_{2}\end{matrix} \right\rbrack} + {\begin{bmatrix}\hat{b} \\0\end{bmatrix}u} + {\begin{bmatrix}l_{1} \\l_{2}\end{bmatrix}\left( {{\overset{.}{y}}_{m} - z_{1}} \right)}}}}\mspace{79mu} {\hat{f} = {\left\lbrack {0\mspace{14mu} 1} \right\rbrack z}}}} & {{Equation}\mspace{14mu} (29)}\end{matrix}$

The shadowed blocks show how the gain matrices of the RESO are obtainedfrom the augmented plant state space model. Equivalently, the RESO canbe represented as

$\begin{matrix}\left\{ {{{\begin{matrix}{\overset{.}{z} = {{A_{z}z} + {B_{z}u} + {L\left( {{\overset{.}{y}}_{m} - {C_{z}z}} \right)}}} \\{\hat{f} = {Vz}}\end{matrix}{where}A_{z}} = \begin{bmatrix}0 & 1 \\0 & 0\end{bmatrix}},{B_{z} = \begin{bmatrix}\hat{b} \\0\end{bmatrix}},{L = \begin{bmatrix}l_{1} \\l_{2}\end{bmatrix}},{C_{z} = \left\lbrack {1\mspace{14mu} 0} \right\rbrack},{V = \left\lbrack {0\mspace{14mu} 1} \right\rbrack},{{{and}\mspace{14mu} l_{1}} = {2\omega_{o}}},{l_{2} = {\omega_{o}^{2}.}}} \right. & {{Equation}\mspace{14mu} (30)}\end{matrix}$

With RESO replacing the ESO 204, the corresponding linear PD control law206 in ADRC 400 is

u=(g _(pd)(e,ė,ω _(c))+{umlaut over (r)}−z ₂)/{circumflex over(b)}  Equation (31)

where

e=r−y _(m) ,ė={dot over (r)}−z ₁  =Equation (32)

If {dot over (y)}_(m) is clean it can be also fed back into the controllaw instead of z₁, the Equation (33) changes to

e=r−y _(m) ,ė={dot over (r)}−{dot over (y)} _(m)  Equation (34)

Note here that the parameterized linear PD control law 206 can bereplaced by the parameterized

NPD control law 506, which will be introduced later.

The order of ESO 204 is reduced further if ÿ is also given, as thedouble derivative of y_(m). According to U.S. patent application Ser.No. 10/351,664, the total disturbance is obtained as x₃=f=ÿ−bu and it issusceptible to measurement noise. In an alternative embodiment, a firstorder RESO can be constructed with the only state z tracking x₃=f. Thecorrespondence between the augmented plant state space model and the newRESO is shown as follows.

$\overset{.}{x} = {{\begin{bmatrix}0 & 1 & 0 \\0 & 0 & 1 \\0 & 0 & 0\end{bmatrix}x} + {\begin{bmatrix}0 \\b \\0\end{bmatrix}u} + {\begin{bmatrix}0 \\0 \\1\end{bmatrix}\overset{.}{f}}}$ y = [1  0  0]x$\overset{.}{z} = {{\lbrack 0\rbrack z} + {\lbrack 0\rbrack u} + {l\left( {{\overset{¨}{y}}_{m} - {\hat{b}u} - z} \right)}}$f̂ = z

The shadowed blocks show how the gain matrices of the RESO are obtainedfrom the augmented plant state space model. Equivalently, the RESO canbe represented as

$\begin{matrix}\left\{ \begin{matrix}{\overset{.}{z} = {{A_{z}z} + {B_{z}u} + {L\left( {{\overset{¨}{y}}_{m} - {\hat{b}u} - {C_{z}z}} \right)}}} \\{\hat{f} = {Vz}}\end{matrix} \right. & {{Equation}\mspace{14mu} (35)}\end{matrix}$

where A_(z)=0, B_(z)=0, L=l=ω_(o), C_(z)=1, V=1. The correspondinglinear PD control law 206 with this RESO is

u=(g _(pd)(e,ė,ω _(c))+{umlaut over (r)}−z)/{circumflex over(b)}  Equation (36)

where e=r−y_(m), ė={dot over (r)}−{dot over (y)}_(m). Note here againthat the parameterized linear PD control law 206 can be replaced by theparameterized NPD control law 506 to be introduced.

A New ESO Implementation

Consider the same second-order plant 202 as described in Equation (9).The ESO 204 described in U.S. patent application Ser. No. 10/351,664 canbe divided into two parts.

$\begin{matrix}\left\{ \begin{matrix}{\begin{bmatrix}{\overset{.}{z}}_{1} \\{\overset{.}{z}}_{2}\end{bmatrix} = {{\begin{bmatrix}0 & 1 \\0 & 0\end{bmatrix}\begin{bmatrix}{\overset{.}{z}}_{1} \\{\overset{.}{z}}_{2}\end{bmatrix}} + {\begin{bmatrix}0 \\\hat{b}\end{bmatrix}u} + {\begin{bmatrix}l_{1} \\l_{2}\end{bmatrix}\left( {y_{m} - z_{1}} \right)}}} \\{z_{3} = {\int{{l_{3}\left( {y_{m} - z_{1}} \right)}{t}}}}\end{matrix} \right. & {{Equation}\mspace{14mu} (37)}\end{matrix}$

The first part is a Luenberger observer for the double integral plant inEquation (13), and the second part shows that the estimation of thegeneral disturbance can be obtained by integrating the observer errorwith a gain of l₃. This implementation is equivalent to the ESO 204.

When implemented in ZOH current estimator form, the Luenberger observeris

$\begin{matrix}\left\{ {{{\begin{matrix}{{z\left\lbrack {k + 1} \right\rbrack} = {{\Phi \; {z\lbrack k\rbrack}} + {\Gamma \; {u\lbrack k\rbrack}} + {L_{p}\left( {{y_{m}\lbrack k\rbrack} - {z_{1}\lbrack k\rbrack}} \right)}}} \\{{\overset{\_}{z}\lbrack k\rbrack} = {{z\lbrack k\rbrack} + {L_{c}\left( {{y_{m}\lbrack k\rbrack} - {z_{1}\lbrack k\rbrack}} \right)}}}\end{matrix}{where}\Phi} = \begin{bmatrix}1 & T \\0 & 1\end{bmatrix}},{\Gamma = \begin{bmatrix}{\hat{b}\frac{T^{2}}{2}} \\{\hat{b}T}\end{bmatrix}},{L_{c} = \begin{bmatrix}{1 - \beta_{2}} \\{\left( {1 + \beta_{2} - {2\beta_{1}}} \right)\text{/}T}\end{bmatrix}},{L_{p} = \begin{bmatrix}{2 - {2\beta_{1}}} \\{\left( {1 + \beta_{1} - {2\beta_{1}}} \right)\text{/}T}\end{bmatrix}},{{{and}\beta_{1}} = {^{\frac{- 3}{2}\omega_{o}T}{\cos \left( {\frac{\sqrt{3}}{2}\omega_{o}T} \right)}}},{\beta_{2} = {^{{- 3}\; \omega_{o}T}.}}} \right. & {{Equation}\mspace{14mu} (38)}\end{matrix}$

Similarly, the RESO in Equation (30) can be divided into two parts:

$\begin{matrix}\left\{ \begin{matrix}{{\overset{.}{z}}_{1} = {{\hat{b}u} + {l_{1}\left( {{\overset{.}{y}}_{m} - z_{1}} \right)}}} \\{z_{2} = {\int{{l_{2}\left( {{\overset{.}{y}}_{m} - z_{1}} \right)}{t}}}}\end{matrix} \right. & {{Equation}\mspace{14mu} (39)}\end{matrix}$

where the first part is a first order Luenberger observer, and thesecond part is an integrator. When implemented in ZOH current estimatorform, this Luenberger observer is

$\begin{matrix}\left\{ \begin{matrix}{{z_{1}\left\lbrack {k + 1} \right\rbrack} = {{\Phi \; {z_{1}\lbrack k\rbrack}} + {\Gamma \; {u\lbrack k\rbrack}} + {L_{p}\left( {{{\overset{.}{y}}_{m}\lbrack k\rbrack} - {z_{1}\lbrack k\rbrack}} \right)}}} \\{{{\overset{\_}{z}}_{1}\lbrack k\rbrack} = {{z_{1}\lbrack k\rbrack} + {L_{c}\left( {{{\overset{.}{y}}_{m}\lbrack k\rbrack} - {z_{1}\lbrack k\rbrack}} \right)}}}\end{matrix} \right. & {{Equation}\mspace{14mu} (40)}\end{matrix}$

where Φ=1, Γ={circumflex over (b)}T, L_(c)=L_(p)=1−β, and β=e^(−2ω) ^(o)^(T).

Prediction of {circumflex over (f)}

With the new implementation of ESO 204, {circumflex over(f)}=∫l₃(y_(m)−z₁)dt, {dot over ({circumflex over (f)})}=l₃(y_(m)−z₁)and {umlaut over ({circumflex over (f)})}=l₃({dot over (y)}_(m)−z₂), andthis enables us to determine {circumflex over (f)}(t+l_(f)) based on the1^(st) or 2^(nd)-order Taylor series approximation shown in

$\begin{matrix}{\mspace{79mu} {{{\hat{f}\left( {t + l_{f}} \right)} \approx {{\int_{\;}^{\;}{{l_{3}\left( {y_{m} - z_{1}} \right)}\ {t}}} + {l_{f}{l_{3}\left( {y_{m} - z_{1}} \right)}}}}\mspace{20mu} {or}}} & {{Equation}\mspace{14mu} (41)} \\{{\hat{f}\left( {t + l_{f}} \right)} \approx {{\int_{\;}^{\;}{{l_{3}\left( {y_{m} - z_{1}} \right)}\ {t}}} + {l_{f}{l_{3}\left( {y_{m} - z_{1}} \right)}} + {\frac{l_{f}^{2}l_{3}}{2}\left( {{\overset{.}{y}}_{m} - z_{2}} \right)}}} & {{Equation}\mspace{14mu} (42)}\end{matrix}$

With RESO, {circumflex over (f)}(t+l_(f)) is obtained as

{circumflex over (f)}(t+l _(f))≈l ₂({dot over (y)} _(m) −z ₁)dt+l _(f) l₂({dot over (y)} _(m) −z ₁)  Equation (43)

Equivalent 2dof Transfer Functions with Prediction in ESO

Using prediction in ESO 204, with the 1^(st)-order Taylor seriesapproximation of future value of {circumflex over (f)} used in controllaw 404, the equivalent 2dof transfer functions are

$\begin{matrix}{{C(s)} = {\frac{1}{\hat{b}}\frac{\begin{matrix}{{l_{f}l_{3}s^{3}} + {\left( {{k_{p}l_{1}} + {k_{d}l_{2}} + l_{3}} \right)s^{3}} +} \\{{\left( {{k_{p}l_{2}} + {k_{d}l_{3}}} \right)s} + {k_{1}l_{3}}}\end{matrix}}{s^{3} + {\left( {l_{1} + k_{d}} \right)s^{2}} + {\left( {l_{2} + k_{p} + {k_{d}l_{1}} - {l_{f}l_{3}}} \right)s}}}} & {{Equation}\mspace{14mu} (44)} \\{{H(s)} = \frac{\left( {s^{3} + {l_{1}s^{2}} + {l_{2}s} + l_{3}} \right)\left( {s^{2} + {k_{d}s} + k_{p}} \right)}{\begin{matrix}{{l_{f}l_{3}s^{3}} + {\left( {{k_{p}l_{1}} + {k_{d}l_{2}} + l_{3}} \right)s^{2}} +} \\{{\left( {{k_{p}l_{2}} + {k_{d}l_{3}}} \right)s} + {k_{1}l_{3}}}\end{matrix}}} & {{Equation}\mspace{14mu} (45)}\end{matrix}$

where l₁, l₂, l₃ are observer gains defined in Equation (11) and k_(p)and k_(d) are the proportional-derivative (PD) gains defined in Equation(17).

When the 2^(nd)-order Taylor series approximation of {circumflex over(f)} is used in control law 404, the equivalent 2dof transfer functionsare

$\begin{matrix}{{C(s)} = {\frac{1}{\hat{b}}\frac{\begin{matrix}{{\frac{l_{f}^{2}l_{3}}{2}s^{4}} + {\left( {{l_{f}l_{3}} + \frac{l_{f}^{2}l_{1}l_{3}}{2}} \right)s^{3}} +} \\{{\left( {{k_{p}l_{1}} + {k_{d}l_{2}} + l_{3}} \right)s^{2}} + {\left( {{k_{p}l_{2}} + {k_{d}l_{3}}} \right)s} + {k_{1}l_{3}}}\end{matrix}}{\begin{matrix}{s^{3} + {\left( {l_{1} + k_{d} - \frac{l_{f}^{2}l_{3}}{2}} \right)s^{2}} +} \\{\left( {l_{2} + k_{p} + {k_{d}l_{1}} - {l_{f}l_{3}} - \frac{l_{f}^{2}l_{1}l_{3}}{2}} \right)s}\end{matrix}}}} & {{Equation}\mspace{14mu} (46)} \\{{H(s)} = \frac{\left( {s^{3} + {l_{1}s^{2}} + {l_{2}s} + l_{3}} \right)\left( {s^{2} + {k_{d}s} + k_{p}} \right)}{\begin{matrix}{{\frac{l_{f}^{2}l_{3}}{2}s^{4}} + {\left( {{l_{f}l_{3}} + \frac{l_{f}^{2}l_{1}l_{3}}{2}} \right)s^{3}} +} \\{{\left( {{k_{p}l_{1}} + {k_{d}l_{2}} + l_{3}} \right)s^{2}} + {\left( {{k_{p}l_{2}} + {k_{d}l_{3}}} \right)s} + {k_{1}l_{3}}}\end{matrix}}} & {{Equation}\mspace{14mu} (47)}\end{matrix}$

Equivalent 2dof Transfer function with Prediction in RESO

The equivalent 2dof transfer functions of ADRC-RESO with prediction in{circumflex over (f)} are as follows.

$\begin{matrix}{{G_{c}(s)} = {\frac{1}{\hat{b}}\frac{{l_{f}l_{2}s^{3}} + {\left( {k_{p} + {k_{d}l_{1}} + l_{2}} \right)s^{2}} + {\left( {{k_{p}l_{1}} + {k_{d}l_{2}}} \right)s} + {k_{p}l_{2}}}{s^{2} + {\left( {k_{d} + l_{1} - {l_{f}l_{2}}} \right)s}}}} & {{Equation}\mspace{14mu} (48)} \\{{H(s)} = \frac{\left( {s^{2} + {l_{1}s} + l_{2}} \right)\left( {k_{p} + {k_{d}s} + s^{2}} \right)}{{l_{f}l_{2}s^{3}} + {\left( {k_{p} + {k_{d}l_{1}} + l_{2}} \right)s^{2}} + {\left( {{k_{p}l_{1}} + {k_{d}l_{2}}} \right)s} + {k_{p}l_{2}}}} & {{Equation}\mspace{14mu} (49)}\end{matrix}$

where l₁,l₂ are observer gains defined in Equation (35).

State Predictor Method for Predictive Extended State Observer

Another embodiment for forming a predictive state and disturbanceobserver 304 module is to use a state predictor method to extend theperformance of a baseline ESO 204 to predict state estimates τseconds inthe future. The basic state predictor method approach was published byT. Oguchi, H. Nijmeijer, Prediction of Chaotic Behavior, IEEE Trans. onCircuits and Systems—I, Vol. 52, No. 11, pp. 2464-2472, 2005, which ishereby incorporated by reference. The state predictor method is combinedin the embodiments presented herein with the regular ESO 204 of the form

$\begin{matrix}{\mspace{79mu} {{\overset{.}{z} = {{Az} + {Bu} + {L\left( {y - \hat{y}} \right)}}}\mspace{79mu} {\hat{y} = {Cz}}\mspace{20mu} {where}}} & {{Equation}\mspace{14mu} (50)} \\{{A = \begin{bmatrix}0 & 1 & 0 \\0 & 0 & 1 \\0 & 0 & 0\end{bmatrix}},{B = \begin{bmatrix}0 \\\hat{b} \\0\end{bmatrix}},{C = \left\lbrack \begin{matrix}1 & 0 & {\left. 0 \right\rbrack,{L = \begin{bmatrix}{3\omega_{o}} \\{3\omega_{o}^{2}} \\\omega_{o}^{3}\end{bmatrix}}}\end{matrix} \right.}} & {{Equation}\mspace{14mu} (51)}\end{matrix}$

to form the predictive ESO is in the form of

{dot over (z)}(t)=Az(t)+Bu(t)+L(y(t)−{circumflex over (y)}(t))

{circumflex over (y)}(t)=Cz(t−τ)  Equation (52)

Then ŷ(t)=z₁(t−τ) will track y(t) while z₁(t)=ŷ(t+τ) is its predictionti seconds ahead. The ideas is that since the observer design ensuresthat the observer error goes to zero, or becomes very small, if thaterror is defined as the difference between y(t) and z₁ (t−τ) and z₁(t−τ)will approach y(t) and z₁(t) will approach y(t+τ). Therefore z₁(t) canbe used as a prediction of y(t) τ second ahead.

Building a Predictive State and Disturbance Observer Using an OutputPrediction Mechanism

Perhaps a simpler method to obtain z(t+τ) is to leave the observerunchanged but replace its input y_(m)(t) with the predicted outputŷ(t+τ) as shown in FIG. 5. This configuration is used both to obtain thepredicted error and its derivatives as well as the predicted disturbanceestimation. Also depicted in FIG. 5 is a prediction module computercomponent or more generally a prediction module 508. The predictionmodule 508 is adapted to accept the measured system output (y_(m)(t))and calculated estimated values of the system output (ŷ(t+τ)) from thepresent time to a time (τ) in the future.

A state and disturbance observer 504, shown in FIG. 5, accepts thepredicted and current values of the plant output (y_(m)) and the controlcommands (u(t)) to estimate the future estimated states (z_(n)(t+τ))where (n) is the number of estimated states plus an estimate of thegeneralized disturbance (f_(n)). The state and disturbance observer 504in one embodiment is an ESO 204. In a second embodiment, the state anddisturbance observer 504 is a disturbance observer (DOB) structure. Inyet another embodiment, the state and disturbance observer 504 is anunknown input observer (UID). It is apparent that the combination of thestate and disturbance observer 504 in combination with the predictionmodule 508 provides a similar output and performance a similar functionwithin the framework of an output predictive ADRC controller 500 as thepredictive state and disturbance observer 304 of the predictive ADRCcontroller 300. In this manner, the various embodiments of thepredictive ADRC controller 300 and output predictive ADRC controller 500are correlated.

Also shown in FIG. 5, a control law 506 is provided that uses theestimated future states of the plant 110 coupled with the desiredtrajectory (r^((n))(t+τ)) to generate a current control input (u(t)).The embodiment of the control law 506 depicted in FIG. 5 incorporatesthe additive inverse of the estimated future values of the estimateddisturbance (f_(n)).

Combine the Output Prediction with Regular ESO

To compensate for the delay, y(t) is replaced with the approximatelypredicted output ŷ(t+τ) as the input to the ESO 204. Again, theapproximated prediction, ŷ(t+τ), is obtained from the Taylor Seriesapproximation. The 1^(st)-order Taylor series approximation and the2^(nd)-order Taylor series approximation of ŷ(t+τ) are as shown in thefollowing equations respectively.

$\begin{matrix}{{\hat{y}\left( {t + \tau} \right)} = {{y_{m}(t)} + {{{\overset{.}{y}}_{m}(t)}\tau}}} & {{Equation}\mspace{14mu} (53)} \\{{\hat{y}\left( {t + \tau} \right)} = {{y_{m}(t)} + {{{\overset{.}{y}}_{m}(t)}\tau} + {\frac{{\overset{¨}{y}}_{m}(t)}{2}\tau^{2}}}} & {{Equation}\mspace{14mu} (54)}\end{matrix}$

Then the predicted system output and the current control signal are usedas the inputs to the ESO 204 as follows.

$\begin{matrix}\left\{ \begin{matrix}{{\overset{.}{z}\left( {t + \tau} \right)} = {{A_{z}{z\left( {t + \tau} \right)}} + {B_{z}{u(t)}} + {L\left( {{\hat{y}\left( {t + \tau} \right)} - {C_{z}{z\left( {t + \tau} \right)}}} \right)}}} \\{{\hat{f}\left( {t + \tau} \right)} = {{Vz}\left( {t + \tau} \right)}}\end{matrix} \right. & {{Equation}\mspace{14mu} (55)}\end{matrix}$

which will provide both the predicted states and the disturbanceestimation in z(t+τ).

Equivalent 2dof Transfer Functions

When the 1^(st)-order Taylor series approximation is used to obtainŷ(t+τ), the equivalent 2dof transfer functions are

$\begin{matrix}{{C(s)} = {\frac{1}{\hat{b}}\frac{\begin{pmatrix}{{\left( {l_{3} + {k_{d}l_{2}} + {l_{1}k_{p}}} \right)s^{2}} +} \\{{\left( {{k_{p}l_{2}} + {l_{3}k_{d}}} \right)s} + {k_{p}l_{3}}}\end{pmatrix}\left( {1 + {\tau \; s}} \right)}{s\left( {s^{2} + {\left( {l_{1} + k_{d}} \right)s} + \left( {l_{2} + k_{p} + {k_{d}l_{1}}} \right)} \right)}}} & {{Equation}\mspace{14mu} (56)} \\{{H(s)} = \frac{\left( {s^{3} + {l_{1}s^{2}} + {l_{2}s} + l_{3}} \right)\left( {k_{p} + {k_{d}s} + s^{2}} \right)}{\begin{pmatrix}{{\left( {l_{3} + {k_{d}l_{2}} + {l_{1}k_{p}}} \right)s^{2}} +} \\{{\left( {{k_{p}l_{2}} + {l_{3}k_{d}}} \right)s} + {k_{p}l_{3}}}\end{pmatrix}\left( {1 + {\tau \; s}} \right)}} & {{Equation}\mspace{14mu} (57)}\end{matrix}$

When the 2^(nd)-order Taylor Series approximation is used to obtainŷ(t+τ), the equivalent 2dof transfer functions are

$\begin{matrix}{{C(s)} = {\frac{1}{\hat{b}}\frac{\begin{pmatrix}{{\left( {l_{3} + {k_{d}l_{2}} + {l_{1}k_{p}}} \right)s^{2}} +} \\{{\left( {{k_{p}l_{2}} + {l_{3}k_{d}}} \right)s} + {k_{p}l_{3}}}\end{pmatrix}\left( {1 + {\tau \; s} + {\frac{\tau^{2}}{2}s^{2}}} \right)}{s\left( {s^{2} + {\left( {l_{1} + k_{d}} \right)s} + \left( {l_{2} + k_{p} + {k_{d}l_{1}}} \right)} \right)}}} & {{Equation}\mspace{14mu} (58)} \\{{H(s)} = \frac{\left( {s^{3} + {l_{1}s^{2}} + {l_{2}s} + l_{3}} \right)\left( {k_{p} + {k_{d}s} + s^{2}} \right)}{\begin{pmatrix}{{\left( {l_{3} + {k_{d}l_{2}} + {l_{1}k_{p}}} \right)s^{2}} +} \\{{\left( {{k_{p}l_{2}} + {l_{3}k_{d}}} \right)s} + {k_{p}l_{3}}}\end{pmatrix}\left( {1 + {\tau \; s} + {\frac{\tau^{2}}{2}s^{2}}} \right)}} & {{Equation}\mspace{14mu} (59)}\end{matrix}$

Combine the Output Prediction with RESOWith the RESO configuration, the prediction is obtained as

{dot over ({circumflex over (y)})}(t+τ)={dot over (y)} _(m)(t)+ŷ_(m)(t)τ  Equation (60)

and the new RESO is constructed as

$\begin{matrix}\left\{ {{{\begin{matrix}{{\overset{.}{z}\left( {t + \tau} \right)} = {{A_{z}{z\left( {t + \tau} \right)}} + {B_{z}{u(t)}} + {L\left( {{\hat{\overset{.}{y}}\left( {t + \tau} \right)} - {C_{z}{z\left( {t + \tau} \right)}}} \right)}}} \\{{\hat{f}\left( {t + \tau} \right)} = {{Vz}\left( {t + \tau} \right)}}\end{matrix}\mspace{20mu} {where}\mspace{20mu} A_{z}} = \begin{bmatrix}0 & 1 \\0 & 0\end{bmatrix}},{B_{z} = \begin{bmatrix}\hat{b} \\0\end{bmatrix}},{L = \begin{bmatrix}l_{1} \\l_{2}\end{bmatrix}},\mspace{20mu} {C_{z} = \left\lbrack {1\mspace{25mu} 0} \right\rbrack},{V = \left\lbrack {0\mspace{25mu} 1} \right\rbrack},\mspace{20mu} {{{and}\mspace{20mu} l_{1}} = {2\omega_{o}}},{l_{2} = {\omega_{o}^{2}.}}} \right. & {{Equation}\mspace{14mu} (61)}\end{matrix}$

which again provides the predicted state and disturbance estimation inz(t+τ).

Equivalent 2 dof Transfer Functions

With the above the 1^(st)-order Taylor Series approximation {dot over(ŷ)}(t+τ) is used as the input to the RESO, the equivalent 2dof transferfunctions are:

$\begin{matrix}{{C(s)} = {\frac{1}{\hat{b}}\frac{\left( {{\left( {k_{p} + {k_{d}l_{1}} + l_{2}} \right)s^{2}} + {\left( {{k_{p}l_{1}} + {l_{2}k_{d}}} \right)s} + {k_{p}l_{2}}} \right)\left( {1 + {\tau \; s}} \right)}{s\left( {s + \left( {l_{1} + k_{d}} \right)} \right)}}} & {{Equation}\mspace{14mu} (62)} \\{{H(s)} = \frac{\left( {s^{2} + {l_{1}s} + l_{2}} \right)\left( {k_{p} + {k_{d}s} + s^{2}} \right)}{\left( {{\left( {k_{p} + {k_{d}l_{1}} + l_{2}} \right)s^{2}} + {\left( {{k_{p}l_{1}} + {l_{2}k_{d}}} \right)s} + {k_{p}l_{2}}} \right)\left( {1 + {\tau \; s}} \right)}} & {{Equation}\mspace{14mu} (63)}\end{matrix}$

Combine the Output Prediction with RESOWith the RESO configuration, the prediction is obtained as

{dot over ({circumflex over (y)})}(t+τ)={dot over (y)} _(m)(t)+ÿ_(m)(t)τ  Equation (64)

and the new RESO is constructed as

$\begin{matrix}\left\{ {{{\begin{matrix}{{\overset{.}{z}\left( {t + \tau} \right)} = {{A_{z}{z\left( {t + \tau} \right)}} + {B_{z}{u(t)}} + {L\left( {{\hat{\overset{.}{y}}\left( {t + \tau} \right)} - {C_{z}{z\left( {t + \tau} \right)}}} \right)}}} \\{{\hat{f}\left( {t + \tau} \right)} = {{Vz}\left( {t + \tau} \right)}}\end{matrix}\mspace{20mu} {where}\mspace{20mu} A_{z}} = \begin{bmatrix}0 & 1 \\0 & 0\end{bmatrix}},{B_{z} = \begin{bmatrix}\hat{b} \\0\end{bmatrix}},{L = \begin{bmatrix}l_{1} \\l_{2}\end{bmatrix}},\mspace{20mu} {C_{z} = \left\lbrack {1\mspace{25mu} 0} \right\rbrack},{V = \left\lbrack {0\mspace{25mu} 1} \right\rbrack},\mspace{20mu} {{{and}\mspace{20mu} l_{1}} = {2\omega_{o}}},{l_{2} = {\omega_{o}^{2}.}}} \right. & {{Equation}\mspace{14mu} (65)}\end{matrix}$

which again provides the predicted state and disturbance estimation inz(t+τ).

Equivalent 2 dof Transfer Functions

With the above the 1^(st)-order Taylor Series approximation {dot over(ŷ)}(t+τ) is used as the input to the RESO, the equivalent 2dof transferfunctions are

$\begin{matrix}{{C(s)} = {\frac{1}{\hat{b}}\frac{\left( {{\left( {k_{p} + {k_{d}l_{1}} + l_{2}} \right)s^{2}} + {\left( {{k_{p}l_{1}} + {l_{2}k_{d}}} \right)s} + {k_{p}l_{2}}} \right)\left( {1 + {\tau \; s}} \right)}{s\left( {s + \left( {l_{1} + k_{d}} \right)} \right)}}} & {{Equation}\mspace{14mu} (66)} \\{{H(s)} = \frac{\left( {s^{2} + {l_{1}s} + l_{2}} \right)\left( {k_{p} + {k_{d}s} + s^{2}} \right)}{\left( {{\left( {k_{p} + {k_{d}l_{1}} + l_{2}} \right)s^{2}} + {\left( {{k_{p}l_{1}} + {l_{2}k_{d}}} \right)s} + {k_{p}l_{2}}} \right)\left( {1 + {\tau \; s}} \right)}} & {{Equation}\mspace{14mu} (67)}\end{matrix}$

A Parameterized Nonlinear PD Control Law, Replacing the Linear PDController

The parameterized linear PD control law 206 g_(pd)(e,ė,ω_(c)) used abovecan be replaced by the following parameterized Nonlinear PD (NPD)control law 506 g_(pd)(e,ė,ω_(c)):

$\begin{matrix}{{y = {{\omega_{c}^{2}e} + {\omega_{c}\overset{.}{e}}}},{x = \left\{ {{\begin{matrix}{{{\omega_{c}\overset{.}{e}} + {\frac{\sqrt{R\left( {R + {8{y}}} \right)} - R}{2}{{sign}(y)}}},} & {{y} > R} \\{{\omega_{c}\left( {{\omega_{c}e} + {2\overset{.}{e}}} \right)},} & {{y} \leq R}\end{matrix}{g_{npd}\left( {e,\overset{.}{e},\omega_{c}} \right)}} = \left\{ {{\begin{matrix}{{R\mspace{14mu} {{sign}(x)}},} & {{x} > R} \\{x,} & {{y} \leq R}\end{matrix}{where}{{sign}(x)}} = \left\{ {\begin{matrix}{1,} & {x \geq 0} \\{{- 1},} & {x < 0}\end{matrix}.} \right.} \right.} \right.}} & {{Equation}\mspace{14mu} (68)}\end{matrix}$

Equation (68) shows that, within the region |x|≦R and |y|≦R, thisparameterized NPD control law 506 is ω_(c) ²e+2ω_(c)ė, which is the sameas the parameterized linear PD control law 206 in Equation (15); outsidethis region, however, the NPD control law 506 is nonlinear. This new NPDcontrol law 506 can be used to replace the linear PD control law 206 inthe original ADRC 200 algorithm to achieve better performance. In oneembodiment the NPD control law 506 is used in conjunction with apredicted state estimate z_(n)(t+τ). In a second embodiment, the NPDcontrol law 506 is uses only the current state estimate (z_(n)(t).

A General Form of Reduced-Order Multiple Extended-State ESO for ann^(th)-Order Plant

Although the Predictive ADRC 300 has been presented with an ESO 204 orRESO with only one extended state, the same technique can be readilyextended to a general n^(th)-order plant with an ESO 240 or RESO thathas multiple extended states. To facilitate such development, we presenthere a general form of the reduced-order multiple extended states ESO204 for an n^(th)-order plant of the form

y ^((n)) =f(y,y, . . . ,y ^((n−1)) ,d,t)+bu  Equation (69)

Let the n states be x₁=y, x₂={dot over (x)}₁={dot over (y)}, . . . ,x_(n)={dot over (x)}_(n−1)=y^((n−1)), and the h extended states bex_(n+1)=f, x_(n+2)=x_(n+1)=f, . . . , x_(+h)={dot over(x)}_(n+h−1)=f^((h−1)). The plant 110 in Equation (69) can now beexpressed in the augmented form:

$\overset{.}{x} = {\begin{bmatrix}{\overset{.}{x}}_{1} \\{\overset{.}{x}}_{2} \\\vdots \\{\overset{.}{x}}_{n + h - m} \\{\overset{.}{x}}_{n + h - m + 1} \\\vdots \\{\overset{.}{x}}_{n} \\{\overset{.}{x}}_{n + 1} \\\vdots \\{\overset{.}{x}}_{n + h}\end{bmatrix} = {\quad{\begin{bmatrix}0 & 1 & 0 & 0 & 0 & \ldots & 0 & 0 & 0 & 0 \\0 & 0 & 1 & 0 & \ddots & \ddots & \ddots & \ddots & \ddots & 0 \\\vdots & \ddots & \ddots & \ddots & 0 & \ddots & \ddots & \ddots & \ddots & \vdots \\0 & 0 & \ddots & 0 & 1 & 0 & \ddots & \ddots & \ddots & 0 \\0 & \ddots & \ddots & \ddots & 0 & 1 & 0 & \ddots & \ddots & 0 \\\vdots & \ddots & \ddots & \ddots & 0 & 0 & 1 & \ddots & \ddots & \vdots \\0 & \ddots & \ddots & \ddots & 0 & 0 & 0 & 1 & \ddots & 0 \\0 & \ddots & \ddots & \ddots & \ddots & \ddots & \ddots & \ddots & \ddots & 0 \\\vdots & \ddots & \ddots & \ddots & \ddots & \ddots & \ddots & \ddots & \ddots & 1 \\0 & 0 & 0 & 0 & 0 & 0 & 0 & \ldots & 0 & 0\end{bmatrix}{\quad{\begin{bmatrix}x_{1} \\x_{2} \\\vdots \\x_{n + h - m} \\x_{n + h - m + 1} \\\vdots \\x_{n} \\x_{n + 1} \\\vdots \\x_{n + h}\end{bmatrix} + {\begin{bmatrix}0 \\0 \\\vdots \\\vdots \\\vdots \\0 \\b \\0 \\\vdots \\0\end{bmatrix}u} + {\begin{bmatrix}0 \\0 \\\vdots \\0 \\0 \\\vdots \\0 \\\vdots \\0 \\1\end{bmatrix}\overset{.}{f}}}}}}}$

or equivalently

                                Equation  (70)$\left\{ {{{\begin{matrix}{\overset{.}{x} = {{A_{x}x} + {B_{x}u} + {Ef}^{(h)}}} \\{y = {C_{x}x}}\end{matrix}\mspace{14mu} {where}A_{x}} = \begin{bmatrix}0 & 1 & 0 & \ldots & 0 \\0 & 0 & 1 & \ddots & \vdots \\0 & 0 & 0 & \ddots & 0 \\\vdots & \ddots & \ddots & \ddots & 1 \\0 & \ldots & 0 & \ldots & 0\end{bmatrix}_{{({n + h})} \times {({n + h})}}},{B_{x} = \begin{bmatrix}0 \\\vdots \\0 \\b \\0 \\\vdots \\0\end{bmatrix}_{{({n + h})} \times 1}},{E = \begin{bmatrix}0 \\\vdots \\0 \\1\end{bmatrix}_{{({n + h})} \times 1}},{C_{x} = \begin{bmatrix}1 & 0 & \ldots & 0\end{bmatrix}_{1 \times {({n + h})}}},{{and}\mspace{14mu} {the}}} \right.$

nonzero element in B_(x) is in the n^(th) row.

Assume that y, {dot over (y)}, . . . , y^((n+h−m)) are measured orotherwise available as y_(m), {dot over (y)}_(m), . . . , y_(m)^((n+h−m)), then the approximate states x₁, x₂, . . . , x_(n+h−m+1) areavailable (If m=h, the last state can be obtained from its definition:x_(n+1)=f=y_(m) ^((n))−{circumflex over (b)}u). Also assume f^((n)) doesnot change rapidly and can be handled by the observer. The ESO 204 ordercan be reduced to m, where h≦m≦n+h. With x_(n+h−m+1) and u as inputs,and with z₁, . . . , z_(m) estimating x_(n+h−m+1), . . . , x_(n+h)respectively, the general ESO 204 form is

$\overset{.}{z} = {\begin{bmatrix}{\overset{.}{z}}_{1} \\\vdots \\{\overset{.}{z}}_{m - h} \\{\overset{.}{z}}_{m - h + 1} \\\vdots \\{\overset{.}{z}}_{m}\end{bmatrix} = {\begin{bmatrix}0 & 1 & 0 & \ldots & \ldots & 0 \\\vdots & \ddots & 1 & \ddots & \ddots & \vdots \\0 & \ddots & \ddots & \ddots & \ddots & 0 \\0 & \ddots & \ddots & \ddots & \ddots & 0 \\\vdots & \ddots & \ddots & \ddots & \ddots & 1 \\0 & 0 & \ldots & \ldots & \ldots & 0\end{bmatrix}{\quad{{\begin{bmatrix}z_{1} \\\vdots \\z_{m - h} \\z_{m - h + 1} \\\vdots \\z_{m}\end{bmatrix} + {\begin{bmatrix}0 \\\vdots \\\hat{b} \\0 \\\vdots \\0\end{bmatrix}u} + {\begin{bmatrix}l_{1} \\\vdots \\l_{m - h} \\l_{m - h + 1} \\\vdots \\l_{m}\end{bmatrix}\left( {x_{n + h - m + 1} - z_{1}} \right)\mspace{79mu} \hat{f}}} = {\begin{bmatrix}0 & \ldots & 0 & 1 & \ldots & 0\end{bmatrix}z}}}}}$

or in state space form

                                Equation  (71)$\left\{ {{{\begin{matrix}{\overset{.}{z} = {{A_{z}z} + {B_{z}u} + {L\left( {x_{n + h - m + 1} - {C_{z}z}} \right)}}} \\{\hat{f} = {Vz}}\end{matrix}\mspace{14mu} {where}A_{z}} = \begin{bmatrix}0 & 1 & 0 & \ldots & 0 \\0 & 0 & 1 & \ddots & \vdots \\0 & 0 & 0 & \ddots & 0 \\\vdots & \ddots & \ddots & \ddots & 1 \\0 & \ldots & 0 & \ldots & 0\end{bmatrix}_{m \times m}},{L = \begin{bmatrix}l_{1} \\l_{2} \\\vdots \\l_{m}\end{bmatrix}_{m \times 1}},{C_{z} = \begin{bmatrix}1 & 0 & \ldots & 0\end{bmatrix}_{1 \times m}},{{{if}\mspace{14mu} m} = h},{{B_{z} = \lbrack 0\rbrack_{m \times 1}};{{{if}\mspace{14mu} m} > h}},{B_{z} = \begin{bmatrix}0 \\\vdots \\0 \\\hat{b} \\0 \\\vdots \\0\end{bmatrix}_{m \times 1}},{V = \begin{bmatrix}0 \\\vdots \\0 \\1 \\0 \\\vdots \\0\end{bmatrix}_{m \times 1}^{T}}} \right.$

Thus, when m>h the nonzero elements of B_(z) and V are in the (m−h)^(th)and (m−h+1)^(th). column respectively.

For tuning simplicity, the observer gains l₁, . . . , l_(m) are chosenas

${l_{i} = {\begin{pmatrix}m \\i\end{pmatrix}\omega_{o}^{i}}},$

0<i≦m.

$\begin{pmatrix}m \\i\end{pmatrix}$

denotes the Binomial Coefficient

$\frac{m!}{{i!}{\left( {m - i} \right)!}}.$

Also, to make the transfer function derivation easier, define

$l_{0} = {{\begin{pmatrix}m \\0\end{pmatrix}\omega_{o}^{0}} = 1.}$

With the state z_(m−h+1) the ESO 204 to estimate f, a control law,u=(u₀−z_(m−h+1))/{circumflex over (b)}, is applied to reduce the plantto a pure n^(th)-order cascade integral plant y^((n))=u₀, which can beeasily controlled using the parameterized linear PD control law 206 ofthe form

$\begin{matrix}{u_{0} = {{\sum\limits_{i = 1}^{n + 1}\; {k_{i}r^{({i - 1})}}} - {\sum\limits_{i = 1}^{n + h - m}\; {k_{i}y_{m}^{({i - 1})}}} - {\sum\limits_{i = 1}^{m - h + 1}\; {k_{n + h - m + i}z_{i}}}}} & {{Equation}\mspace{14mu} (72)}\end{matrix}$

where

${k_{i} = {\begin{pmatrix}m \\{i - 1}\end{pmatrix}\omega_{c}^{n + 1 - i}}},$

0≦i≦n+1, and the middle term,

${\sum\limits_{i = 1}^{n + h - m}\; {k_{i}y^{({i - 1})}}},$

is corresponding to the order reduction (it disappears when the inputsof ESO 204 are y_(m) and u). In the case where the measured output y_(m)is clean and its derivatives can be obtained from y_(m), ADRC 200 canalso be applied as

$\begin{matrix}{u_{0} = {{\sum\limits_{i = 1}^{n + 1}\; {k_{i}r^{({i - 1})}}} - {\sum\limits_{i = 1}^{n}\; {k_{i}y_{m}^{({i - 1})}}} - z_{m - h + 1}}} & {{Equation}\mspace{14mu} (73)}\end{matrix}$

The corresponding transfer function representation of ADRC 400, in a2dof structure (as shown in FIG. 4) is

$\begin{matrix}{{H(s)} = \frac{\Psi_{0,m}^{m}{\sum\limits_{j = 1}^{n + 1}\; {k_{j}s^{j - 1}}}}{\begin{matrix}{{\Psi_{0,m}^{m}{\sum\limits_{j = 1}^{n + h - m}\; {k_{j}s^{({j - 1})}}}} +} \\{\sum\limits_{j = 1}^{m - h + 1}\; {k_{n + h - m + j}s^{n + h - 1 - m + j}\Psi_{j,m}^{m}}}\end{matrix}}} & {{Equation}\mspace{14mu} (74)} \\{{{G_{c}(s)} = {\frac{1}{\hat{b}}\frac{\begin{matrix}{{\Psi_{0,m}^{m}{\sum\limits_{j = 1}^{n + h - m}\; {k_{j}s^{({j - 1})}}}} +} \\{\sum\limits_{j = 1}^{m - h + 1}\; {k_{n + h - m + j}s^{n + h - 1 - m + j}\Psi_{j,m}^{m}}}\end{matrix}}{\sum\limits_{j = 1}^{m - h + 1}\; {k_{n + h - m + j}s^{h - 1 + j - m}\Psi_{0,{j - 1}}^{m}}}}}{{{{where}\mspace{14mu} \Psi_{x,y}^{m}} = {\sum\limits_{i = x}^{y}\; {l_{i}s^{m - i}}}},{0 \leq x \leq {y.}}}} & {{Equation}\mspace{14mu} (75)}\end{matrix}$

ADRC with Partially Known Plant Dynamics

ADRC controller does not require the detail plant model information.However, if the whole or part of the plant dynamics is known, variousembodiments of the ADRC controller can be modified to utilize theinformation to estimate and reject disturbance more effectively, eitherby control law modification or ESO 204 modification. With a part of theplant dynamics known, the known part of the generalized disturbance canbe generated and rejected in the control law directly, while the rest ofthe generalized disturbance is estimated by ESO 204 and then rejected inthe linear PD control law 206 as previously shown in the original ADRC200 framework. The other way to utilize the partially known plantdynamics is adapting the dynamics into ESO 204, where ESO 204 stillestimates the generalized disturbance.

Consider the second-order plant 202 in Equation (8), with the knowledgeof the plant information, part of the generalized disturbance is known,denoted as f_(n)(y, {dot over (y)}). The rest is unknown part of thegeneralized disturbance, denoted as f_(u)(y,{dot over (y)},d,t). Theplant in Equation (9) can now be described as

ÿ=f _(n)(y,{dot over (y)})+f _(u)(y,{dot over (y)},d,t)+bu  Equation(76)

and there are two ways to take advantage of new information inf_(n)(y,{dot over (y)}) and incorporate it into ADRC 200. One way is toimprove ESO 204 by combining f_(n)(x) with A_(x)x+B_(x)u in the plantspace description in Equation (76). The augmented state space form ofthe plant is

$\begin{matrix}\left\{ \begin{matrix}{\overset{.}{x} = {{A_{x}x} + {E{{\overset{.}{f}}_{n}(x)}} + {B_{x}u} + {E{\overset{.}{f}}_{u}}}} \\{y = {C_{x}x}}\end{matrix} \right. & {{Equation}\mspace{14mu} (77)}\end{matrix}$

and the corresponding ESO 204 is

$\begin{matrix}\left\{ \begin{matrix}{\overset{.}{z} = {{A_{x}z} + {E{{\overset{.}{f}}_{n}(z)}} + {B_{x}u} + {L\left( {y_{m} - {C_{x}z}} \right)}}} \\{\hat{f} = {Vz}}\end{matrix} \right. & {{Equation}\mspace{14mu} (78)}\end{matrix}$

Then a new ESO 204 can be constructed to estimate the generalizeddisturbance f based on the new state space description of the plant 110in Equation (77). For example, consider the plant ÿ=a₁{dot over (y)}+buwhere a₁ is given, i.e. f_(n)(y,{dot over (y)})=a₁{dot over (y)}. Theknown part of the generalized disturbance is obtained from the modelinformation: {dot over (f)}_(n)=−a₁ÿ=−a₁(−a₁{dot over (y)}+bu). In thiscase where f_(n)(y,{dot over (y)}) is a linear function, the augmentedstate space representation of the plant 110 can be written as

$\begin{matrix}\left\{ {\begin{matrix}{\overset{.}{x} = {{{\overset{\_}{A}}_{x}x} + {{\overset{\_}{B}}_{x}u} + {E{\overset{.}{f}}_{u}}}} \\{y = {C_{x}x}}\end{matrix},{{{where}{\overset{\_}{A}}_{x}} = \begin{bmatrix}0 & 1 & 0 \\0 & 0 & 1 \\0 & a_{1}^{2} & 0\end{bmatrix}},{{\overset{\_}{B}}_{x} = \begin{bmatrix}0 \\b \\{{- a_{1}}b}\end{bmatrix}}} \right. & {{Equation}\mspace{14mu} (79)}\end{matrix}$

and the new ESO is ż=Ā_(x)z+B _(z)u+L(y_(m)−C_(x)z), where B _(z)=[0{circumflex over (b)}−a₁{circumflex over (b)}]^(T).

The other way of taking advantage of the additional information inf_(n)(y,{dot over (y)}) is to use it to cancel known plant dynamics inthe control law directly. There are three ways to do this:

u=(g(e,ė,ω _(c))+{umlaut over (r)}−f _(n)(y _(m) ,{dot over (y)} _(m))−z₃)/{circumflex over (b)}  Equation (80)

or y_(m) and {dot over (y)}_(m) can be replaced by z₁ and z₂ in thecontrol law,

u=(g(e,ė,ω _(c))+{umlaut over (r)}−f _(n)(z ₁ ,z ₂)−z ₃)/{circumflexover (b)}  Equation (81)

or y_(m) and {dot over (y)}_(m) can be replaced by r and {dot over (r)}respectively and rejected in the control law in a feedforward manner.

u=(g(e,ė,ω _(c))+{umlaut over (r)}−f _(n)(r,{dot over (r)})−z₃)/{circumflex over (b)}  Equation (82)

where g(e,ė,ω_(c)) is the parameterized PD controller 206 or NPDcontroller defined in Equation (15) or Equation (68) respectively. Ofcourse, any combination of the above three methods are available for useby one of ordinary skill in the art.

INDUSTRIAL APPLICATIONS

The present active disturbance rejection controller, and its associatedprocesses, methods and devices as disclosed herein possesses a number ofunique attributes and industrial applicability, including for example,utility as a controller for tracking control, web processingapplications, and jet engine control applications as described ingreater detail above.

For example, in the case of tracking or motion control applications, thevarious predictive methods presented herein, particularly the ones thatuse the knowledge of the plant dynamics, offer reduced phase lag in theobserver and compensate for the time delay in the plant itself, all ofwhich especially benefit tracking applications that employ a desiredoutput trajectory, sometimes called motion profile.

CONCLUSION

While various embodiments of the present system and method for feedbackcontrol of systems have been described above, it should be understoodthat the embodiments have been presented by the way of example only, andnot limitation. It will be understood by those skilled in the art thatvarious changes in form and details may be made therein withoutdeparting from the spirit and scope of the invention as defined. Thus,the breadth and scope of the present invention should not be limited byany of the above described exemplary embodiments.

What is claimed is:
 1. A system for controlling velocity of a plant,comprising: a first order state space model comprising a velocity value,v(t), where v(t) equals the sum of f(t) and b*u(t), f(t) represents thecombined effects of internal dynamics and external disturbance of theplant, u(t) is a control signal, and b is a constant to an approximatevalue; a linear extended state observer, which is a function of a singleperformance parameter, for estimating the value of f(t); and a monitoradapted to cancel the effect of f(t) on the velocity value by anestimate from the linear extended state observer.
 2. The system of claim1, where the plant is a web processing system.
 3. The system of claim 2,where the velocity value comprises at least one of a carriage rollervelocity, an exit-side roller velocity, and a process-side rollervelocity.
 4. The system of claim 1, where the linear extended stateobserver comprises a reduced order extended state observer.
 5. Thesystem of claim 2, where the web processing system comprises a webhandling machine for processing paper.
 6. The system of claim 2, wherethe web processing system comprises a web handling machine forprocessing plastic film.
 7. The system of claim 2, where the webprocessing system comprises a web handling machine for processing clothfabrics.
 8. The system of claim 2, where the web processing systemcomprises a web handling machine for processing strip steel.
 9. Thesystem of claim 1, where the velocity value v(t) comprises velocities ofrollers at either end of a web span.
 10. The system of claim 1, wherethe combined effects of internal dynamics and external disturbance ofthe plant comprise effects of temperature and humidity.
 11. The systemof claim 1, where the combined effects of internal dynamics and externaldisturbance of the plant comprise effects of machine wear and variationsin raw material.
 12. A control system for a plant, comprising: aplurality of active disturbance rejection controllers; and a highfrequency gain matrix implemented in an electronic device for receivingat least one input signal from each of the plurality of activedisturbance rejection controllers and for producing a plurality ofcontrol signals u, where the plant receives the plurality of controlsignals u and produces a plurality of output signals y, each of saidplurality of output signals further provided as an input to at least oneof the plurality of active disturbance rejection controllers.
 13. Thecontrol system for the plant of claim 12, where the high-frequency gainmatrix is a single gain matrix.
 14. The control system for the plant ofclaim 12, where the high-frequency gain matrix comprises a plurality ofmatrices with a one-to-one association between each input signal andeach control signal.
 15. The control system for the plant of claim 12,where the plant is a turbofan.
 16. The control system for the plant ofclaim 12, where the plant is one of a chemical process, a mechanicalprocess, or an electrical process.
 17. The control system for a plant ofclaim 12, where the plant is one of a chemical process, an airplaneflight control, a missile flight control, a CNC machine control, arobotics control, a magnetic bearing, a satellite attitude control, or aprocess control.